On π-electron configuration of cyclopenta-derivatives of benzenoid hydrocarbons

Size: px
Start display at page:

Download "On π-electron configuration of cyclopenta-derivatives of benzenoid hydrocarbons"

Transcription

1 Indian Journal of Chemistry Vol. 49, July 010, pp On π-electron configuration of cyclopenta-derivatives of benzenoid hydrocarbons Ivan Gutman*, Jelena Đurđević, Dušan Bašić & Dragana Rašović Faculty of Science, P.O. Box 60, Kragujevac, Serbia Received 19 May 010; accepted 15 June 010 Within the Hückel molecular orbital model, Kekulèan benzenoid hydrocarbons have equal number of bonding and antibonding molecular orbitals, and no non-bonding molecular orbitals. In cyclopenta-derivatives of benzenoid hydrocarbons this regularity may be either preserved or violated, depending on the number and position of the fivemembered rings. Some general regularities along these lines are established. In particular, the problem is completely solved in the case of prolate-rectangle-shaped benzenoids; the number of bonding molecular orbitals always exceeds the number of antibonding molecular orbitals, except if all five-membered rings lie on the same side of the benzenoid system. Keywords: Theoretical chemistry, Graph theory, Electron configuration, Molecular orbitals, Hydrocarbons, Benzenoids In an attempt to extend the earlier well developed theory of benzenoid hydrocarbons 1-5 to their derivatives containing five-membered rings, we have recently initiated a systematic study of the π-electron properties of such species containing one 6-14 or two cyclopentadiene fragments. lready in the case of dicyclopenta-derivatives, we encountered pronounced irregularities 15,16 caused by the fact that these have more bonding than antibonding molecular orbitals (MOs), and therefore some of their bonding MOs are unoccupied. The natural question that arises at this point is what happens in the general case when the benzenoid derivatives possess more than one fivemembered ring. The present work reports our findings on the π-electron configuration of cyclopentaderivatives of benzenoid hydrocarbons, possessing two or more five-membered rings. It is worth noting that some preliminary researches along these lines were undertaken as early as in the 1970s. 19 Some background theories required to deduce the main results herein are reviewed prior to discussion on the findings in the present study. Theoretical Chemical graph theory The chemical graph theory is well documented in literature. 0,1 In this section we mention some results needed for the subsequent considerations. Let G be a (molecular) graph, possessing n vertices. Let ( G ) be the adjacency matrix of G, and let λ1 λ λn be the eigenvalues of ( G ), referred to as the eigenvalues of the graph G. s is well known, 0- these eigenvalues are closely related to the Hückel molecular orbital (HMO) energy levels of the underlying conjugated π-electron system. Let n +, n, and n 0 be the number of eigenvalues of G that are, respectively, positive, negative, and equal to zero. In HMO theory, n +, n, and n 0 are the number of bonding, antibonding and non-bonding MOs. The characteristic polynomial of the graph G is defined as φ( G, λ) = det[ λ I ( G)] where I is the unit matrix of order n. Then, φ ( G,0) = det[ ( G)] = ( 1) det ( G). On the other hand, det ( G) = λ1 λ λ3 λn. Let G be a graph, and e its edge connecting the vertices u and v. Let Z1, Z,, Zt be the cycles of G in which the edge e is contained. Then the characteristic polynomial of G satisfies the following recurrence relationship, 3-5 φ( G, λ) = φ( G e, λ) φ( G u v, λ) φ( G Z, λ) n t i= 1 i (1) graph is said to be bipartite if its vertices can be coloured by two colours (say, black and white), so that the colour of adjacent vertices is always different.

2 854 INDIN J CHEM, SEC, JULY 010 In Fig. 1 are depicted two bipartite graphs and the colouring of their vertices indicated. graph is bipartite if and only if it does not possess odd-membered cycles. Therefore, the molecular graphs of benzenoid hydrocarbons are bipartite, but those of their cyclopenta-derivatives are not. For bipartite graphs, according to the pairing theorem, λi = λ n + 1 i holds for all i = 1,,, n. Thus, for bipartite graphs, n + = n -. Let G be a bipartite graph, and let n b of its vertices be coloured black and n w coloured white, nb + nw = n. Then CE = CE( G) = nb nw is called the colour excess of G. For example, for the graphs depicted in Fig. 1, CE( G 1) = 0 and CE( G ) =. It is known that n 0 CE, implying that if the colour excess is non-zero, then there exists at least one zero eigenvalue. Then the product of all eigenvalues is equal to zero. Consequently, we arrive at a later used result, Lemma 1: If a bipartite graph G has non-zero colour excess, then det ( G ) = 0 and φ ( G,0) = 0. Topological theory of benzenoid hydrocarbons The fundamentals of the theory of benzenoid molecules is detailed in literature. Here we re-state some of their properties needed for the subsequent considerations. We denote by K( G ) the number of Kekulè structures of the molecule whose molecular graph is G. With respect to K, benzenoid systems are classified as Kekulèan (with K > 0 ) and non-kekulèan (with K = 0 ). It is easy to see that the colour excess of Kekulèan benzenoids is zero and vice versa a benzenoid system with colour excess greater than zero must be non-kekulèan. For instance, one of the benzenoid systems depicted in Fig. 1 has zero colour excess and is Kekulèan, K( G 1) = 9, whereas the other has non-zero colour excess and is non-kekulèan, K(G ) = 0. The Dewar-Longuet-Higgins formula,6 establishes an algebraic relation between the eigenvalues of a benzenoid system and its Kekulè structure count, n/ det ( G) = ( 1) K( G) which, combined with the pairing theorem, yields K( G) = λ λ λn. 1 / Fig. 1 Two benzenoid systems with coloured vertices. In the theory of benzenoid systems it is customary that the peaks are coloured white and the valleys black. The graph G 1 has 1 black and 1 white vertices, and therefore its colour excess is zero. The graph G has 13 black and 11 white vertices and its colour excess is. For what follows we need a special case of the above results, namely, Lemma : If G is a Kekulèan benzenoid system, then the number n of its vertices is even, n+ = n = n /, and n 0 = 0. In other words, within the HMO approximation, a Kekulèan benzenoid system has equal number of bonding and antibonding MOs and has no non-bonding MO. Results and Discussion Monocyclopenta-derivatives of benzenoid hydrocarbons Theorem 3: If B is any Kekulèan benzenoid system, and H is its any monocyclopenta-derivative, then n+ ( H ) = n ( H ) and n0 ( H ) = 0, i.e., H has equal number of bonding and antibonding MOs, and has no non-bonding MO. Proof: Let B be a Kekulèan benzenoid system, and let its vertices be coloured in the above described manner (for example see Fig. ). Then the fivemembered ring of H is attached to B either via two white or via two black vertices. Without loss of generality we assume that the five-membered ring is attached to two white vertices p and q (cf. Fig. ). The two newly added vertices of H may be labelled as u and v, and the edge between them by e (cf. Fig. ). Further, let H(k) be the graph in which the edges of H, connecting the vertices p and u, and q and v have weights k (cf. Fig. ). This means that for k = 1 the graph H(k) coincides with H, whereas for k = 0 the graph H(k) consists of two disjoint components: B and a two-vertex path (cf. Fig. ). In view of the structure of H(0), the set of its eigenvalues is the union of the eigenvalues of B and { 1, + 1}. Therefore, n ( H (0)) = n ( H (0)) and n 0 ( H (0) = 0.

3 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 855 Fig. Kekulèan benzenoid system B, its cyclopenta-derivative H and the auxiliary graphs used for proving Theorem 3. For details see text. pplying Eq. (1) to H(k) we obtain, φ( H ( k), λ) = φ( H ( k) e, λ) φ( H ( k) u v, λ) t k φ( H ( k) Zi, λ) i= 1 () We have now to observe that all the graphs occurring on the right-hand side of Eq. () are bipartite. In particular, (a) H ( k) e is the graph obtained from B by attaching pendent vertices to its vertices p and q. Since both p and q are white, the two new vertices must be coloured black. Consequencely, the colour excess of H ( k) e is equal to two. Then, according to Lemma 1, φ( H ( k) e,0) = 0. (b) H ( k) u v is just the benzenoid system B. (c) The subgraph H ( k) Zi may be viewed as being obtained by deleting from B a path starting at vertex p and ending at vertex q. Since p and q have equal colours, this path contains an odd number of vertices. Therefore H ( k) Zi is a bipartite graph with odd number of vertices. Then, according to Lemma 1, φ( H ( k) Z i,0) = 0 for all i = 1,,, t. Bearing the above in mind, setting λ = 0 into Eq. (), and recalling Lemma, we get n/ φ( H( k),0) = φ( B,0) = det ( B) = ( 1) K( B) 0 Thus, the product of the eigenvalues of H(k) is independent of k, and is non-zero. This means that if the parameter k monotonically increases from k = 0 to k = 1, no eigenvalue of H(k) will change sign. Since n ( H (0)) = n ( H (0)) and n 0 ( H (0) = 0 it must be n+ ( H ( k)) = n ( H ( k)) and n0 ( H ( k ) = 0 for any value of k. Then, for k = 1 we arrive at the claim of Theorem 3. Dicyclopenta-derivatives of benzenoid hydrocarbons In the case of dicyclopenta-derivatives of benzenoid hydrocarbons we must distinguish between two cases. We refer to them as syn and anti. Two cyclopentadiene fragments are in syn position, if both five-membered rings are attached to vertices of the same colour of the parent benzenoid system. Two cyclopentadiene fragments are in anti position, if the two five-membered rings are attached to vertices of different colour of the parent benzenoid system. For an illustrative example see Fig. 3. Theorem 4: If B is any Kekulèan benzenoid system, and S is its any syn-dicyclopenta-derivative,

4 856 INDIN J CHEM, SEC, JULY 010 Fig. 4 Vertices, edges and five-membered rings of S. Notation used in the proofs of Theorems 4 and 5. Fig. 3 G 3 is a syn-dicyclopenta- and G 4 an anti-dicyclopentaderivative of a benzenoid system G 5. The colouring of vertices of G 5 is indicated. then n+ ( S) = n ( S) and n0 ( S ) = 0, i. e., S has equal number of bonding and antibonding MOs, and has no non-bonding MO. Proof is similar to that of Theorem 3. The vertices, edges, and five-membered rings of S are labelled as indicated in Fig. 4. Let S(k) be obtained by assigning a weight k to the edges pu, vq, rx, and sy of the graph S. Then a twofold application of Eq. (1) yields, 4 φ( S( k), λ) = α β k γ k δ where α = φ( S( k) e f, λ) φ( S( k) e x y, λ) k φ( S( k) e ZB, λ) B k φ( S( k) u v ZB, λ) B (3) β = φ( S( k) u v f, λ) φ( S( k) u v x y, λ) γ (4) φ ( S( k) Z f, λ) φ( S( k) Z x y, λ) = k φ( S( k) Z ZB, λ) B (5) δ = φ( S( k) Z, λ) (6) In the above formulas, and B indicate summation over cycles that, respectively embrace the ring but not the ring B, embrace the ring B but not the ring, and embrace both rings and B. In the case syn-sicyclopenta-derivatives, the situation is much simplified since all subgraphs occurring in Eqs. (3)-(6) are bipartite and, with the exception of S( k) u v x y, have colour excess greater than zero. In particular, CE( S( k) e f ) = 4, CE( S( k) e x y) =, CE( S( k) e Z B ) = 3, CE( S( k) u v f ) =. CE( S( k) u v Z B ) = 1, CE( S( k) Z f ) = 3, CE( S( k) Z x y) = 1, CE( S( k) Z ZB ) =, and CE( S( k) Z ) =. Therefore, settingλ = 0, noting that S( k) u v x y B, and by applying Lemmas 1 and, we get φ n/ ( S( k),0) = φ( B,0) = det ( B) = ( 1) K( B) 0 This implies that the product of the eigenvalues of S(k) is independent of k, and is non-zero. Theorem 4 follows now in a fully analogous manner as Theorem 3. Theorem 4 remains valid also if there are more than five-membered rings, provided all of them are in syn-constellation. The case of anti-dicyclopenta-derivatives is significantly more complicated, and for such polycyclic conjugated systems the equality of number of bonding and antibonding MOs may be violated. 15,16,19 From a mathematical point of view, the difficulties arise because several bipartite subgraphs occurring in Eqs. (3)-(6) have now zero colour excess. Theorem 5: Let B be a Kekulèan benzenoid system, and R be its any anti-dicyclopenta-derivative. Then in

5 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 857 the notation specified above and in Fig. 4, n ( R) > n ( R) will hold provided the inequality K( B) + K( B p q r s) < B 4 K( R Z Z ) + K( R Z ) B (7) is satisfied. In other words, if the above inequality holds, then R has more bonding than antibonding MOs. The proof for the above will be only sketched. φ( R, λ ) is expanded in the same way as before (cf. Eqs (3)-(6)) and from φ ( R,0) the terms with nonzero colour excess are eliminated. Then, to the remaining terms the Dewar-Longuet-Higgins formula is applied, noting that the number of vertices of R is four greater than the number of vertices of B. Finally, it can be proven 7 that all pairs of cycles Z, Z for which K( R Z ZB ) 0 have a total number of vertices divisible by four, and this also holds for the cycles Z for which K( R Z ) 0. This leads to the conclusion that if the number n of vertices of B is divisible by 4, then the product of the eigenvalues of R is equal to K( B) + K( B p q r s) 4 K( R Z ZB ) K( R Z ) B whereas the product of eigenvalues of B is equal to K( B ) > 0. If n is even, but not divisible by four, then the product of the eigenvalues of R is equal to B Prolate-rectangle-shaped benzenoid hydrocarbons: case study It is evident that in order that inequality (7) be satisfied, K(B) should be small. One class of Kekulèan benzenoid systems with relatively small number of Kekulè structure are the so-called prolate rectangles. 8 Their general structure, denoted by R( n, m), n, m 1, is depicted in Fig. 5. The anti-dicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1 have numerous pairs of disjoint odd-membered cycles Z, Z B, see Fig. 6. Furthermore, some of the subgraphs obtained by deleting these cycles have large Kekulè structure counts, causing the value of 4 K( R Z ZB ) B in (7) to significantly exceed the value of K( B) + K( B p q r s). For instance, in the case of the anti-dicyclopenta-derivative of R(4,) depicted in Fig. 6, K( B) + K( B p q r s) = = 81 which is much smaller than 4 K( R Z ZB ) = B 4( =... > 3000 fully analogous case is encountered at all antidicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1, enabling one to formulate: Rule 6: ll anti-dicyclopenta-derivatives of prolate rectangles R(n,m) with m > 1 have n+ > n, that is, have more bonding than antibonding MOs. K R Z ZB + K R Z B 4 ( ) ( ) K( B) K( B p q r s) whereas the product of eigenvalues of B is equal to K( B) < 0. In both cases, the validity of Eq. (7) implies that the parity of the number of positive eigenvalues of B and R is different. Since n+ ( B) = n ( B), this implies n+ ( R) n ( R). Knowing that the presence of five-membered rings can only increase the number of bonding MOs, 19 we arrive at n ( R) > n ( R). Fig. 5 The prolate rectangle R(n,m), possessing mn + (m 1)(n 1) hexagons and (n+1) m Kekulè structures. 8

6 858 INDIN J CHEM, SEC, JULY 010 Fig. 6 Some, among the more than one hundred pairs, of the disjoint odd-membered cycles Z, Z B of an anti-dicyclopenta-derivative of R(4,). The Kekulè structure counts of the cycle-deleted subgraphs, pertaining to diagrams 1,,, 8, are 9, 14, 14, 14, 9, 0, 1 and 0, respectively. (For completeness, recall that all syn-dicyclopentaderivatives of prolate rectangles R(n,m) with m > 1 have n+ = n, that is, have equal number of bonding and antibonding MOs.) Corroborated by our extensive numerical calculations, the above rule can be extended to arbitrary cyclopenta-derivatived of R(n,m): Rule 7: cyclopenta-derivative of prolate rectangles R(n,m) with m > 1 has n+ > n, and thus more bonding than antibonding MOs, if and only if it possesses at least one pair of five-membered rings in anti-constellation. Otherwise, n = n. The case of prolate rectangles R(n,m) for m = 1, namely the case of linear polyacenes, is exceptional (cf. Fig. 7). If m = 1, then the arguments leading to Rule 6 do not apply, and also Rule 6 does not generally hold. By numerical testing we found that there exist anti-dicyclopenta-derivatives of R( n,1) Ln, for which n+ = n, i. e., which have equal number of bonding and antibonding MOs. The smallest such molecules are depicted in Fig. 7. For larger values of n, such violations of Rule 6 are more numerous. For instance, n+ = n holds for L 1 (1,8), L 1 (1,9), L 1 (1,10), L 1 (1,11), and L 1 (,10). Evidently, the condition for the equality of the numbers of bonding and antibonding MOs is that the two cyclopentadiene fragments are sufficiently far from each other. The numerical data obtained for n-values up to 15 indicate that n+ = n holds for Ln (1, t ) whenever t ( n 1) / 3 + ( n 1) / 3 + 1, which remains a challenge for to be mathematically verified for the general case.

7 GUTMN et al.: π -ELECTRON CONFIGURTION OF CYCLOPENT-DERIVTIVES OF BENZENOID HYDROCRBONS 859 Fig. 7 The linear polyacene Ln R( n,1) and the labelling of its anti-dicyclopenta-derivatives. L 4 (1,3), L 5 (1,4), L 6 (1,4), and L 6 (1,5), and are the smallest such molecules having equal number of bonding and antibonding MO, thus violating Rule 6 for m = 1. Conclusions The present study was aimed at revealing the structural conditions required by the cyclopentaderivatives of benzenoid hydrocarbons to have equal number of bonding and antibonding Hückel molecular orbitals. It has been recognized 19,9-31 that this is a necessary condition that the respective molecule has a stable π-electron configuration, and thus be chemically stable. This was also confirmed by our recent studies on dicyclopenta-species. It is hoped that the general results reported in this paper will shed more light on this problem, and in addition, show how the mathematical apparatus of chemical graph theory could be used for the treatment of problems of this kind. cknowledgement The authors thank the support by the Serbian Ministry of Science, through grant no G, and project "Graph Theory and Mathematical Programming with pplications to Chemistry and Engineering". References 1 Clar E, The romatic Sextet (Wiley, London) 197. Gutman I & Cyvin S J, Introduction to the Theory of Benzenoid Hydrocarbons (Springer-Verlag, Berlin) dvances in the Theory of Benzenoid Hydrocarbons, edited by I Gutman & S J Cyvin, (Springer-Verlag, Berlin) dvances in the Theory of Benzenoid Hydrocarbons II, edited by I Gutman (Springer-Verlag, Berlin) Randić M, Chem Rev 103 (003) Gutman I, & Đurđević J, MTCH Commun Math Comput Chem, 60 (008) Gutman I, Đurđević J, & Balaban T, Polyc rom Comp, 9 (009) 3. 8 Đurđević J, Gutman I, Terzić J & Balaban T, Polyc rom, Comp, 9 (009) Balaban T, Đurđević J & Gutman I, Polyc rom Comp, 9 (009) Gutman I, Đurđević J, Radenković, S & Burmudžija, Indian J Chem, 48 (009) Gutman I, Jeremić S & Petrović V, Indian J Chem, 48 (009) Marković, S, Stanković S, Radenković S &. Gutman I, Monat Chem, 140 (009) Đurđević J, Radenković, S, Gutman I & Marković S, Monat Chem, 140 (009) Balaban T, Dickens T K, Gutman I & Mallion R B, Croat Chem cta, 8 (010), (In press). 15 Gutman I & Furtula B, Polyc rom Comp, 8 (008) Gutman I, Đurđević J, Furtula B & Milivojević B, Indian J Chem, 47 (008) Marković S, Stanković S, Radenković S & Gutman I, J Chem Inf Model, 48 (008) Stanković S, Marković S, Radenković S & Gutman I, J Mol Model, 15 (009) 953.

8 860 INDIN J CHEM, SEC, JULY Gutman I, Trinajstić N & Živković T, Tetrahedron, 9 (1973) Trinajstić N, Chemical Graph Theory, (CRC Press, Boca Raton) Gutman I & Polansky O E, Mathematical Concepts in Organic Chemistry, (Springer-Verlag, Berlin) Dias J R, Molecular Orbital Calculations Using Chemical Graph Theory, (Springer-Verlag, Berlin) Heilbronner E, Helv Chim cta, 36 (1953) Schwenk J, in Graphs and Combinatorics, edited by R Bari & F Harary, (Springer-Verlag, Berlin) 1974, pp Gutman I & Polansky O E, Theor Chim cta, 60 (1981) Dewar M J S & Longuet-Higgins H C, Proc Royal Soc London, 14 (195) Gutman I & Đurđević J, MTCH Commun Math Comput Chem, 65 (011) Cyvin S J & Gutman I, Kekulè Structure in Benzenoid Hydrocarbons, (Springer-Verlag, Berlin) 1988, pp Streitwieser, Molecular Orbital Theory for Organic Chemists, (Wiley, New York) Dewar M J S, The Molecular Orbital Theory of Organic Chemistry, (McGraw-Hill, New York) Dewar M J S & Dougherty R C, The PMO Theory of Organic Chemistry, (Plenum Press, New York) 1975.

Available online at

Available online at J. Serb. Chem. Soc. 75 (1) 83 90 (2010) UDC 547.678.004.14+547.534:530.18:66.022.362 JSCS 3943 Original scientific paper Effect of a ring on the cyclic conjugation in another ring: applications to acenaphthylene-type

More information

EFFECT OF BENZOCYCLOBUTADIENO-ANNELATION ON CYCLIC CONJUGATION IN PERYLENE. Ivan Gutman and Beba Stojanovska

EFFECT OF BENZOCYCLOBUTADIENO-ANNELATION ON CYCLIC CONJUGATION IN PERYLENE. Ivan Gutman and Beba Stojanovska Macedonian Journal of Chemistry and Chemical Engineering, Vol. 30, No. 2, pp. 235 240 (2011) MJCCA9 585 ISSN 1857-5552 Received: March 4, 2011 UDC: 547.6 Accepted: May 5, 2011 Original scientific paper

More information

Available online at

Available online at J. Serb. Chem. Soc. 76 (5) 733 741 (2011) UDC 547.686:54.02+66.095.252.5:547.52 JSCS 4154 Original scientific paper Effect of benzocyclobutadieno-annelation on cyclic conjugation in fluoranthene congeners

More information

Calculating the hyper Wiener index of benzenoid hydrocarbons

Calculating the hyper Wiener index of benzenoid hydrocarbons Calculating the hyper Wiener index of benzenoid hydrocarbons Petra Žigert1, Sandi Klavžar 1 and Ivan Gutman 2 1 Department of Mathematics, PEF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia

More information

ALGEBRAIC STRUCTURE COUNT OF ANGULAR HEXAGONAL-SQUARE CHAINS

ALGEBRAIC STRUCTURE COUNT OF ANGULAR HEXAGONAL-SQUARE CHAINS ALGEBRAIC STRUCTURE COUNT OF ANGULAR HEXAGONAL-SQUARE CHAINS Olga Bodroža-Pantić Department of Mathematics and Informatics, Faculty of Sciences, Trg D. Obradovića 4, University of Novi Sad, 21000 Novi

More information

Relating resonance energy with the Zhang Zhang polynomial

Relating resonance energy with the Zhang Zhang polynomial J. Serb. Chem. Soc. 72 (7) 665 671 (2007) UDC 547.537:541.6+512.62:539.194 JSCS 3599 Original scientific paper Relating resonance energy with the Zhang Zhang polynomial SABINA GOJAK 1, IVAN GUTMAN 2 *

More information

Algebraic structure count of linear phenylenes and their congeners *

Algebraic structure count of linear phenylenes and their congeners * J.Serb.Chem.Soc. 68(4 5)391 399(2003) UDC 517.986.9:547.77 JSCS 3054 Original scientific paper Algebraic structure count of linear phenylenes and their congeners * IVAN GUTMAN # Faculty of Science, University

More information

FIBONACCI NUMBERS AND ALGEBRAIC STRUCTURE COUNT OF SOME NON-BENZENOID CONJUGATED POLYMERS

FIBONACCI NUMBERS AND ALGEBRAIC STRUCTURE COUNT OF SOME NON-BENZENOID CONJUGATED POLYMERS FIBONACCI NUMBERS AND ALGEBRAIC STRUCTURE COUNT OF SOME NON-BENZENOID CONJUGATED POLYMERS Olga Bodroza-Pantid Institute of Mathematics, University of Novi Sad, Novi Sad, Yugoslavia Ivan Gutmaii Faculty

More information

Clar number of catacondensed benzenoid hydrocarbons

Clar number of catacondensed benzenoid hydrocarbons Clar number of catacondensed benzenoid hydrocarbons Sandi Klavžar a,, Petra Žigert a, Ivan Gutman b, a Department of Mathematics, PEF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia b Faculty

More information

Normal components, Kekulé patterns, and Clar patterns in plane bipartite graphs

Normal components, Kekulé patterns, and Clar patterns in plane bipartite graphs Normal components, Kekulé patterns, and Clar patterns in plane bipartite graphs Wai Chee Shiu a, Peter Che Bor Lam a, Fuji Zhang b, Heping Zhang c a Department of Mathematics, Hong Kong Baptist University,

More information

Electron content of rings of fully benzenoid hydrocarbons

Electron content of rings of fully benzenoid hydrocarbons J. Serb. Chem. Soc. 70 (10) 1199 1204 (2005) UDC 547.53:537.12 JSCS 3357 Original scientific paper Electron content of rings of fully benzenoid hydrocarbons IVAN GUTMAN 1,*#, BORIS FURTULA 1, SVETLANA

More information

On benzenoid systems with a minimal number of inlets

On benzenoid systems with a minimal number of inlets J. Serb. Chem. Soc. 78 (9) 1351 1357 (2013) UDC 547.53+547.535:544.112: JSCS 4502 548.1.023:548.115 Original scientific paper On benzenoid systems with a minimal number of inlets ROBERTO CRUZ 1, IVAN GUTMAN

More information

The maximum forcing number of a polyomino

The maximum forcing number of a polyomino AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 69(3) (2017), Pages 306 314 The maximum forcing number of a polyomino Yuqing Lin Mujiangshan Wang School of Electrical Engineering and Computer Science The

More information

On the sextet polynomial of fullerenes

On the sextet polynomial of fullerenes On the sextet polynomial of fullerenes Jean-Sébastien Sereni Matěj Stehlík Abstract We show that the sextet pattern count of every fullerene is strictly smaller than the Kekulé structure count. This proves

More information

WHAT CHEMISTS COULD NOT SEE WITHOUT MATHEMATICS Dependence of total π-electron energy on molecular structure

WHAT CHEMISTS COULD NOT SEE WITHOUT MATHEMATICS Dependence of total π-electron energy on molecular structure Xjenza 2005; 0, p. 3-7 3 Invited Article WHAT CHEMIT COULD NOT EE WITHOUT MATHEMATIC Dependence of total -electron energy on molecular structure Ivan Gutman Faculty of cience, University of Kragujevac,

More information

NOTE ON THE COULSON AND COULSON JACOBS INTEGRAL FORMULAS

NOTE ON THE COULSON AND COULSON JACOBS INTEGRAL FORMULAS MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 59 (2008) 257-268 ISSN 0340-6253 NOTE ON THE COULSON AND COULSON JACOBS INTEGRAL FORMULAS Miodrag Mateljević

More information

Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 26

Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 26 Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts - 001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 6 SOME SPECTRAL PROPERTIES OF STARLIKE TREES M. LEPOVIĆ, I.

More information

Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 2013, pp On terminal Wiener indices of kenograms and plerograms

Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 2013, pp On terminal Wiener indices of kenograms and plerograms Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 013, pp. 77 89 IJMC On terminal Wiener indices of kenograms and plerograms I. GUTMAN a,, B. FURTULA a, J. TOŠOVIĆ a, M. ESSALIH b AND M. EL

More information

What is the meaning of the graph energy after all?

What is the meaning of the graph energy after all? What is the meaning of the graph energy after all? Ernesto Estrada and Michele Benzi Department of Mathematics & Statistics, University of Strathclyde, 6 Richmond Street, Glasgow GXQ, UK Department of

More information

WHAT CHEMISTS COULD NOT SEE WITHOUT MATHEMATICS - DEPENDENCE OF TOTAL π-electron ENERGY ON MOLECULAR STRUCTURE

WHAT CHEMISTS COULD NOT SEE WITHOUT MATHEMATICS - DEPENDENCE OF TOTAL π-electron ENERGY ON MOLECULAR STRUCTURE 57 Kragujevac J. ci. 27 (2005) 57-66. WHAT CHEMIT COULD NOT EE WITHOUT MATHEMATIC - DEPENDENCE OF TOTAL π-electron ENERGY ON MOLECULAR TRUCTURE Ivan Gutman Faculty of cience, P. O. Box 60, 34000 Kragujevac,

More information

2009 Copyright (CC) SCS

2009 Copyright (CC) SCS J. Serb. Chem. Soc. 74 (5) 549 554 (2009) UDC 547.534+547.629:66.022.362+537.872 JSCS 3854 Original scientific paper Verifying the PCP-rule by five-center bond indices JELENA ĐURĐEVIĆ 1, IVAN GUTMAN 1

More information

The Hall rule in fluoranthene-type benzenoid hydrocarbons

The Hall rule in fluoranthene-type benzenoid hydrocarbons J. Serb. Chem. Soc. 73 (10) 989 995 (2008) UDC 537.872+539.124:547.686:54.022+541.5 JSCS 3780 Original scientific paper The Hall rule in fluoranthene-type benzenoid hydrocarbons JELENA ĐURĐEVIĆ, SLAVKO

More information

Energy of a polynomial and the Coulson integral formula

Energy of a polynomial and the Coulson integral formula J Math Chem (21) 48:162 168 DOI 1.17/s191-1-9725-z ORIGINAL PAPER Energy of a polynomial and the Coulson integral formula Miodrag Mateljević Vladimir Božin Ivan Gutman Received: 3 November 29 / Accepted:

More information

D-EQUIENERGETIC SELF-COMPLEMENTARY GRAPHS

D-EQUIENERGETIC SELF-COMPLEMENTARY GRAPHS 123 Kragujevac J. Math. 32 (2009) 123 131. D-EQUIENERGETIC SELF-COMPLEMENTARY GRAPHS Gopalapillai Indulal 1 and Ivan Gutman 2 1 Department of Mathematics, St. Aloysius College, Edathua, Alappuzha 689573,

More information

Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and the Same Wiener Index

Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and the Same Wiener Index MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 78 (2017) 121-132 ISSN 0340-6253 Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and

More information

Partitioning of π-electrons in rings of aza-derivatives of naphthalene

Partitioning of π-electrons in rings of aza-derivatives of naphthalene J. Serb. Chem. Soc. 72 (7) 655 663 (2007) UDC 54.724:547.7+547.831+547.652:534.24 JSCS 3598 Original scientific paper Partitioning of π-electrons in rings of aza-derivatives of naphthalene IVAN GUTMAN*

More information

ON EQUIENERGETIC GRAPHS AND MOLECULAR GRAPHS

ON EQUIENERGETIC GRAPHS AND MOLECULAR GRAPHS Kragujevac J. Sci. 29 2007) 73 84. UDC 541.66:547.318 ON EQUIENERGETIC GRAPHS AND MOLECULAR GRAPHS Hanumappa B. Walikar, a Harishchandra S. Ramane, b Ivan Gutman, c Sabeena B. Halkarni a a Department of

More information

RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS

RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS Kragujevac J. Sci. 30 (2008) 57-64. UDC 541.61 57 RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS Slavko Radenković and Ivan Gutman Faculty of Science, P. O. Box 60, 34000 Kragujevac, Republic of Serbia

More information

Estimation of the HOMO-LUMO Separation

Estimation of the HOMO-LUMO Separation CROATICA CHEMICA ACTA CCACAA 53 (1) 45-50 (1980) CCA-1187 Estimation of the HOMO-LUMO Separation A. Graovac* and I. Gutman** YU ISSN 0011-1643 UDC 539.19 Original Scientific Paper *»Ruder Boskovic«Institute,

More information

LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS

LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS Kragujevac Journal of Mathematics Volume 39() (015), Pages 193 05. LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS HARISHCHANDRA S. RAMANE 1, GOURAMMA A. GUDODAGI 1,

More information

Topological Properties of Benzenoid Systems. IX *. On the Sextet Polynomial

Topological Properties of Benzenoid Systems. IX *. On the Sextet Polynomial Topological Properties of Benzenoid Systems. IX *. On the Sextet Polynomial Ivan Gutman Faculty of Science, University of Kragujevac, Yugoslavia Z. Naturforsch. 37 a, 69 73 (1982); received September 26,

More information

HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS

HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS Kragujevac J. Sci. 28 (2006) 47 56. HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS Hanumappa B. Walikar, a Harishchandra S. Ramane, b Leela Sindagi, a Shailaja S. Shirakol a and Ivan Gutman c

More information

New Approaches for the Real and Complex Integral Formulas of the Energy of a Polynomial

New Approaches for the Real and Complex Integral Formulas of the Energy of a Polynomial MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 66 (0) 849-86 ISSN 0340-653 New Approaches for the Real and Complex Integral Formulas of the Energy of a

More information

THE MATCHING ENERGY OF A GRAPH

THE MATCHING ENERGY OF A GRAPH THE MATCHING ENERGY OF A GRAPH IVAN GUTMAN AND STEPHAN WAGNER Abstract. The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G. We define the matching energy ME of the

More information

On trees with smallest resolvent energies 1

On trees with smallest resolvent energies 1 On trees with smallest resolvent energies 1 Mohammad Ghebleh, Ali Kanso, Dragan Stevanović Faculty of Science, Kuwait University, Safat 13060, Kuwait mamad@sci.kuniv.edu.kw, akanso@sci.kuniv.edu.kw, dragance106@yahoo.com

More information

Elementary Blocks of Plane Bipartite Graphs

Elementary Blocks of Plane Bipartite Graphs Elementary Blocks of Plane Bipartite Graphs Peter C.B. Lam, W.C. Shiu, Heping Zhang Abstract Let G be a 2-connected plane bipartite graph with perfect matchings, with more than one cycle and with minimum

More information

On a Class of Distance Based Molecular Structure Descriptors

On a Class of Distance Based Molecular Structure Descriptors On a Class of Distance Based Molecular Structure Descriptors Franka Miriam Brückler a, Tomislav Došlić b, Ante Graovac c, Ivan Gutman d, a Department of Mathematics, University of Zagreb, Bijenička 30,

More information

RELATION BETWEEN ENERGY AND LAPLACIAN ENERGY

RELATION BETWEEN ENERGY AND LAPLACIAN ENERGY MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 59 (200) 343-354 ISSN 0340-6253 RELATION BETWEEN ENERGY AND LAPLACIAN ENERGY Ivan Gutman, a NairMariaMaiadeAbreu,

More information

Resonance graphs of kinky benzenoid systems are daisy cubes

Resonance graphs of kinky benzenoid systems are daisy cubes arxiv:1710.07501v1 [math.co] 20 Oct 2017 Resonance graphs of kinky benzenoid systems are daisy cubes Petra Žigert Pleteršek Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

More information

CCO Commun. Comb. Optim.

CCO Commun. Comb. Optim. Communications in Combinatorics and Optimization Vol. 3 No., 018 pp.179-194 DOI: 10.049/CCO.018.685.109 CCO Commun. Comb. Optim. Leap Zagreb Indices of Trees and Unicyclic Graphs Zehui Shao 1, Ivan Gutman,

More information

Resolvent Energy of Graphs

Resolvent Energy of Graphs Resolvent Energy of Graphs I.Gutman 1,2, B.Furtula 1, E.Zogić 2, E.Glogić 2 1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia 2 State University of Novi Pazar, Novi Pazar, Serbia May 19,

More information

Enumeration of subtrees of trees

Enumeration of subtrees of trees Enumeration of subtrees of trees Weigen Yan a,b 1 and Yeong-Nan Yeh b a School of Sciences, Jimei University, Xiamen 36101, China b Institute of Mathematics, Academia Sinica, Taipei 1159. Taiwan. Theoretical

More information

ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS

ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2010) 91-100 ISSN 0340-6253 ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER

More information

A Survey on Comparing Zagreb Indices

A Survey on Comparing Zagreb Indices MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (2011) 581-593 ISSN 0340-6253 A Survey on Comparing Zagreb Indices Bolian Liu and Zhifu You School of

More information

COUNTING RELATIONS FOR GENERAL ZAGREB INDICES

COUNTING RELATIONS FOR GENERAL ZAGREB INDICES Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 95 103. COUNTING RELATIONS FOR GENERAL ZAGREB INDICES G. BRITTO ANTONY XAVIER 1, E. SURESH 2, AND I. GUTMAN 3 Abstract. The first and second

More information

Solutions to Unsolved Problems on the Minimal Energies of Two Classes of Graphs

Solutions to Unsolved Problems on the Minimal Energies of Two Classes of Graphs MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 66 (2011) 943-958 ISSN 0340-6253 Solutions to Unsolved Problems on the Minimal Energies of Two Classes of

More information

CHEMICAL GRAPH THEORY

CHEMICAL GRAPH THEORY CHEMICAL GRAPH THEORY SECOND EDITION Nenad Trinajstic, Ph.D. Professor of Chemistry The Rugjer Boskovic Institute Zagreb The Republic of Croatia CRC Press Boca Raton Ann Arbor London Tokyo TABLE OF CONTENTS

More information

A note on the Laplacian Estrada index of trees 1

A note on the Laplacian Estrada index of trees 1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2009) 777-782 ISSN 0340-6253 A note on the Laplacian Estrada index of trees 1 Hanyuan Deng College of

More information

Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons

Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons /;C' // - MODELS IN CHEMISTRY 136 (5-6), pp 539-548 (1999) Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons IVANGUTMAN1, ELENA V. KONSTANTINOVA2 and VLADIMIR A. SKOROBOGATOV2

More information

LAPLACIAN ESTRADA INDEX OF TREES

LAPLACIAN ESTRADA INDEX OF TREES MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (009) 769-776 ISSN 0340-653 LAPLACIAN ESTRADA INDEX OF TREES Aleksandar Ilić a and Bo Zhou b a Faculty

More information

THE EDGE VERSIONS OF THE WIENER INDEX

THE EDGE VERSIONS OF THE WIENER INDEX MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 61 (2009) 663-672 ISSN 0340-6253 THE EDGE VERSIONS OF THE WIENER INDEX Ali Iranmanesh, a Ivan Gutman, b

More information

An Application of the permanent-determinant method: computing the Z-index of trees

An Application of the permanent-determinant method: computing the Z-index of trees Department of Mathematics wtrnumber2013-2 An Application of the permanent-determinant method: computing the Z-index of trees Daryl Deford March 2013 Postal address: Department of Mathematics, Washington

More information

Extremal Values of Vertex Degree Topological Indices Over Hexagonal Systems

Extremal Values of Vertex Degree Topological Indices Over Hexagonal Systems MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (013) 501-51 ISSN 0340-653 Extremal Values of Vertex Degree Topological Indices Over Hexagonal Systems

More information

arxiv: v1 [math.co] 6 Feb 2011

arxiv: v1 [math.co] 6 Feb 2011 arxiv:1102.1144v1 [math.co] 6 Feb 2011 ON SUM OF POWERS OF LAPLACIAN EIGENVALUES AND LAPLACIAN ESTRADA INDEX OF GRAPHS Abstract Bo Zhou Department of Mathematics, South China Normal University, Guangzhou

More information

On the difference between the revised Szeged index and the Wiener index

On the difference between the revised Szeged index and the Wiener index On the difference between the revised Szeged index and the Wiener index Sandi Klavžar a,b,c M J Nadjafi-Arani d June 3, 01 a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia sandiklavzar@fmfuni-ljsi

More information

EQUIENERGETIC GRAPHS

EQUIENERGETIC GRAPHS 5 Kragujevac J. Math. 26 (2004) 5 13. EQUIENERGETIC GRAPHS Harishchandra S. Ramane, 1 Hanumappa B. Walikar, 2 Siddani Bhaskara Rao, 3 B. Devadas Acharya, 4 Prabhakar R. Hampiholi, 1 Sudhir R. Jog, 1 Ivan

More information

Dependence of the total -electron energy on a large number of non-bonding molecular orbitals

Dependence of the total -electron energy on a large number of non-bonding molecular orbitals J. Serb. Chem. Soc. 69 (10) 777 782 (2004) UDC 54 724+537.872:519.17:54 12 JSCS 3204 Original scientific paper Dependence of the total -electron energy on a large number of non-bonding molecular orbitals

More information

arxiv: v1 [math.co] 30 Dec 2015

arxiv: v1 [math.co] 30 Dec 2015 Resolvent Energy of Unicyclic, Bicyclic and Tricyclic Graphs arxiv:1512.08938v1 [math.co] 30 Dec 2015 Luiz Emilio Allem 1, Juliane Capaverde 1, Vilmar Trevisan 1, Abstract Ivan Gutman 2,3, Emir Zogić 3,

More information

Technische Universität Ilmenau Institut für Mathematik

Technische Universität Ilmenau Institut für Mathematik Technische Universität Ilmenau Institut für Mathematik Preprint No. M 05/07 On Maximum Matchings and Eigenvalues of Benzenoid Graphs Fajtlowicz, Siemion; John, Peter E.; Sachs, Horst Mai 2005 Impressum:

More information

Graphs with Integer Matching Polynomial Roots

Graphs with Integer Matching Polynomial Roots Graphs with Integer Matching Polynomial Roots S. Akbari a, P. Csikvári b, A. Ghafari a, S. Khalashi Ghezelahmad c, M. Nahvi a a Department of Mathematical Sciences, Sharif University of Technology, Tehran,

More information

Extremal trees with fixed degree sequence for atom-bond connectivity index

Extremal trees with fixed degree sequence for atom-bond connectivity index Filomat 26:4 2012), 683 688 DOI 10.2298/FIL1204683X Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Extremal trees with fixed degree

More information

CCO Commun. Comb. Optim.

CCO Commun. Comb. Optim. Communications in Combinatorics and Optimization Vol. 1 No. 2, 2016 pp.137-148 DOI: 10.22049/CCO.2016.13574 CCO Commun. Comb. Optim. On trees and the multiplicative sum Zagreb index Mehdi Eliasi and Ali

More information

arxiv: v2 [math.co] 4 Sep 2009

arxiv: v2 [math.co] 4 Sep 2009 Bounds for the Hückel energy of a graph Ebrahim Ghorbani a,b,, Jack H. Koolen c,d,, Jae Young Yang c a Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran,

More information

Verifying the modes of cyclic conjugation in tetrabenzo[bc,ef,op,rs]circumanthracene

Verifying the modes of cyclic conjugation in tetrabenzo[bc,ef,op,rs]circumanthracene J. Serb. Chem. Soc. 77 (10) 1401 1408 (2012) UDC 547.53:66 945.3+544.02+519.677 JSCS 4361 Original scientific paper Verifying the modes of cyclic conjugation in tetrabenzo[bc,ef,op,rs]circumanthracene

More information

Two Mathematical Papers Relevant for the Hückel Molecular Orbital Theory 1

Two Mathematical Papers Relevant for the Hückel Molecular Orbital Theory 1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 72 (2014) 565-572 ISSN 0340-6253 Two Mathematical Papers Relevant for the Hückel Molecular Orbital Theory

More information

A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX

A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX AKBAR ALI 1 AND NENAD TRINAJSTIĆ arxiv:1705.1040v1 [math.co] 0 May 017 Abstract. In the paper [Gutman, I.; Trinajstić, N. Chem. Phys. Lett. 197, 17, 55],

More information

arxiv: v1 [math.co] 26 Nov 2014

arxiv: v1 [math.co] 26 Nov 2014 A Maximum Resonant Set of Polyomino Graphs arxiv:1411.716v1 [math.co] 6 Nov 014 Heping Zhang, Xiangqian Zhou School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China

More information

Ring Currents and the PCP Rule

Ring Currents and the PCP Rule CROATICA CHEMICA ACTA CCACAA, ISSN 0011-1643, e-issn 1334-417X Croat. Chem. Acta 83 (2) (2010) 209 215. CCA-3411 Original Scientific Article Ring Currents and the PCP Rule A. T. Balaban,a T. K. Dickens,b

More information

SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R 1 OF A GRAPH

SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R 1 OF A GRAPH ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number, 207 SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R OF A GRAPH EI MILOVANOVIĆ, PM BEKAKOS, MP BEKAKOS AND IŽ MILOVANOVIĆ ABSTRACT Let G be an undirected

More information

Computing distance moments on graphs with transitive Djoković-Winkler s relation

Computing distance moments on graphs with transitive Djoković-Winkler s relation omputing distance moments on graphs with transitive Djoković-Winkler s relation Sandi Klavžar Faculty of Mathematics and Physics University of Ljubljana, SI-000 Ljubljana, Slovenia and Faculty of Natural

More information

DISC6024 MODC+ ARTICLE IN PRESS. Discrete Mathematics ( ) Note UNCORRECTED PROOF

DISC6024 MODC+ ARTICLE IN PRESS. Discrete Mathematics ( ) Note UNCORRECTED PROOF PROD. TYPE: FTP PP: -6 (col.fig.: nil) DISC6024 MODC+ ED: Smita PAGN: Vish -- SCAN: Global 2 2 2 2 2 Discrete Mathematics ( ) Note www.elsevier.com/locate/disc On the role of hypercubes in the resonance

More information

On the adjacency matrix of a block graph

On the adjacency matrix of a block graph On the adjacency matrix of a block graph R. B. Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit

More information

ENERGY OF SOME CLUSTER GRAPHS

ENERGY OF SOME CLUSTER GRAPHS 51 Kragujevac J. Sci. 23 (2001) 51-62. ENERGY OF SOME CLUSTER GRAPHS H. B. Walikar a and H. S. Ramane b a Karnatak University s Kittur Rani Chennama Post Graduate Centre, Department of Mathematics, Post

More information

The Fibonacci numbers for the molecular graphs of linear phenylenes

The Fibonacci numbers for the molecular graphs of linear phenylenes The Fibonacci numbers for the molecular graphs of linear phenylenes Jaroslav Seibert 1 and Libor Koudela 2 Institute of Mathematics and Quantitative Methods Faculty of Economics and Administration University

More information

Some spectral inequalities for triangle-free regular graphs

Some spectral inequalities for triangle-free regular graphs Filomat 7:8 (13), 1561 1567 DOI 198/FIL138561K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Some spectral inequalities for triangle-free

More information

Estrada Index of Graphs

Estrada Index of Graphs Estrada Index of Graphs Mohammed Kasim 1 Department of Mathematics, University of Kashan, Kashan, Iran Email: kasimmd@kashanu.ac.ir Fumao Zhang, Qiang Wang Department of Mathematics, Xi an University of

More information

A test of Clar aromatic sextet theory

A test of Clar aromatic sextet theory J. Serb. Chem. Soc. 78 (10) 1539 1546 (2013) UDC 541:547.52+547.53+549.746 JSCS 4516 Original scientific paper A test of Clar aromatic sextet theory IVAN GUTMAN* #, SLAVKO RADENKOVIĆ #, MARIJA ANTIĆ and

More information

Journal of Mathematical Nanoscience. Strong chromatic index of certain nanosheets

Journal of Mathematical Nanoscience. Strong chromatic index of certain nanosheets Journal of Mathematical Nanoscienece 7 () (07) 9 8 Journal of Mathematical Nanoscience Available Online at: http://jmathnano.sru.ac.ir Strong chromatic index of certain nanosheets Vidya Ganesan, Indra

More information

The Linear Chain as an Extremal Value of VDB Topological Indices of Polyomino Chains

The Linear Chain as an Extremal Value of VDB Topological Indices of Polyomino Chains Applied Mathematical Sciences, Vol. 8, 2014, no. 103, 5133-5143 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46507 The Linear Chain as an Extremal Value of VDB Topological Indices of

More information

On spectral radius and energy of complete multipartite graphs

On spectral radius and energy of complete multipartite graphs Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn., ISSN 1855-3974 (electronic edn. ARS MATHEMATICA CONTEMPORANEA 9 (2015 109 113 On spectral radius and energy of complete multipartite

More information

Stability of bicalicene isomers A topological study

Stability of bicalicene isomers A topological study J. Serb. Chem. Soc. 81 (1) 81 89 (2016) UDC 547.534:544.131:537.872 12:66 945.3 JSCS 4829 Original Scientific paper Stability of bicalicene isomers A topological study IVAN GUTMAN* # Faculty of Science,

More information

Given any simple graph G = (V, E), not necessarily finite, and a ground set X, a set-indexer

Given any simple graph G = (V, E), not necessarily finite, and a ground set X, a set-indexer Chapter 2 Topogenic Graphs Given any simple graph G = (V, E), not necessarily finite, and a ground set X, a set-indexer of G is an injective set-valued function f : V (G) 2 X such that the induced edge

More information

Discrete Applied Mathematics

Discrete Applied Mathematics iscrete Applied Mathematics 57 009 68 633 Contents lists available at Scienceirect iscrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam On reciprocal complementary Wiener number Bo

More information

Enumeration of perfect matchings of a type of Cartesian products of graphs

Enumeration of perfect matchings of a type of Cartesian products of graphs Discrete Applied Mathematics 154 (006) 145 157 www.elsevier.com/locate/dam Enumeration of perfect matchings of a type of Cartesian products of graphs Weigen Yan a,b,, Fui Zhang b a School of Sciences,

More information

arxiv: v1 [math.co] 5 Sep 2016

arxiv: v1 [math.co] 5 Sep 2016 Ordering Unicyclic Graphs with Respect to F-index Ruhul Amin a, Sk. Md. Abu Nayeem a, a Department of Mathematics, Aliah University, New Town, Kolkata 700 156, India. arxiv:1609.01128v1 [math.co] 5 Sep

More information

HOMO-LUMO MAPS AND GOLDEN GRAPHS. Tomaž Pisanski, Slovenia CSD5, Sheffield, England Wednesday, July 21, 2010

HOMO-LUMO MAPS AND GOLDEN GRAPHS. Tomaž Pisanski, Slovenia CSD5, Sheffield, England Wednesday, July 21, 2010 HOMO-LUMO MAPS AND GOLDEN GRAPHS Tomaž Pisanski, Slovenia CSD5, Sheffield, England Wednesday, July 21, 2010 Outline HOMO-LUMO maps Recently, we introduced a graphical tool for investigating spectral properties

More information

arxiv: v1 [math.co] 15 Sep 2016

arxiv: v1 [math.co] 15 Sep 2016 A Method for Computing the Edge-Hyper-Wiener Index of Partial Cubes and an Algorithm for Benzenoid Systems Niko Tratnik Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia arxiv:1609.0469v1

More information

Zagreb indices of block-edge transformation graphs and their complements

Zagreb indices of block-edge transformation graphs and their complements Indonesian Journal of Combinatorics 1 () (017), 64 77 Zagreb indices of block-edge transformation graphs and their complements Bommanahal Basavanagoud a, Shreekant Patil a a Department of Mathematics,

More information

THE HOSOYA INDEX AND THE MERRIFIELD-SIMMONS INDEX OF SOME GRAPHS. Communicated by Alireza Ashrafi. 1. Introduction

THE HOSOYA INDEX AND THE MERRIFIELD-SIMMONS INDEX OF SOME GRAPHS. Communicated by Alireza Ashrafi. 1. Introduction Transactions on Combinatorics ISSN (print): 51-8657, ISSN (on-line): 51-8665 Vol 1 No (01), pp 51-60 c 01 University of Isfahan wwwcombinatoricsir wwwuiacir THE HOSOYA INDEX AND THE MERRIFIELD-SIMMONS

More information

Some Remarks on the Arithmetic Geometric Index

Some Remarks on the Arithmetic Geometric Index Iranian J. Math. Chem. 9 (2) June (2018) 113 120 Iranian Journal of Mathematical Chemistry n Journal homepage: ijmc.kashanu.ac.ir Some Remarks on the Arithmetic Geometric Index JOSE LUIS PALACIOS Department

More information

JOURNAL OF MATHEMATICAL NANOSCIENCE. Connective eccentric index of fullerenes

JOURNAL OF MATHEMATICAL NANOSCIENCE. Connective eccentric index of fullerenes JOURNAL OF MATHEMATICAL NANOSCIENCE JMNS Vol 1, No 1, 2011, 43-50 Connective eccentric index of fullerenes MODJTABA GHORBANI Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training

More information

Applications of Isometric Embeddings to Chemical Graphs

Applications of Isometric Embeddings to Chemical Graphs DIMACS Series in Discrete Mathematics and Theoretical Computer Science Applications of Isometric Embeddings to Chemical Graphs Sandi Klavžar Abstract. Applications of isometric embeddings of benzenoid

More information

Key words: matching polynomial; acyclic polynomial; matching generating polynomial; perfect matching; Hosoya index; Pfaffian orientation

Key words: matching polynomial; acyclic polynomial; matching generating polynomial; perfect matching; Hosoya index; Pfaffian orientation On the Matching Polynomials of Graphs with Small Number of Cycles of Even Length WEIGEN YAN,, YEONG-NAN YEH, FUJI ZHANG 3 School of Sciences, Jimei University, Xiamen 360, China Institute of Mathematics,

More information

The PI Index of the Deficient Sunflowers Attached with Lollipops

The PI Index of the Deficient Sunflowers Attached with Lollipops Int. J. Contemp. Math. Sciences, Vol. 6, 011, no. 41, 001-007 The PI Index of the Deficient Sunflowers Attached with Lollipops Min Huang Institute of Mathematics Physics and Information Sciences Zhejiang

More information

On the Energy of Some Graphs

On the Energy of Some Graphs nnals of Pure and pplied Mathematics Vol. 7, No., 08, 5- ISSN: 79-087X P, 79-0888online Published on pril 08 www.researchmathsci.org DOI: http://dx.doi.org/0.57/apam.v7na nnals of On the Energy of Some

More information

Graphenes Aromatic Giants

Graphenes Aromatic Giants Ivan Gutman and Boris Furtula Graphenes Aromatic Giants Ivan Gutman and Boris Furtula Ivan Gutman is Professor of Physical Chemistry at the Faculty of Science, Kragujevac, Serbia. Among his interests are

More information

arxiv: v1 [math.co] 13 Mar 2018

arxiv: v1 [math.co] 13 Mar 2018 A note on polyomino chains with extremum general sum-connectivity index Akbar Ali, Tahir Idrees arxiv:1803.04657v1 [math.co] 13 Mar 2018 Knowledge Unit of Science, University of Management & Technology,

More information

Improved bounds on the difference between the Szeged index and the Wiener index of graphs

Improved bounds on the difference between the Szeged index and the Wiener index of graphs Improved bounds on the difference between the Szeged index and the Wiener index of graphs Sandi Klavžar a,b,c M. J. Nadjafi-Arani d October 5, 013 a Faculty of Mathematics and Physics, University of Ljubljana,

More information

SEIDEL ENERGY OF ITERATED LINE GRAPHS OF REGULAR GRAPHS

SEIDEL ENERGY OF ITERATED LINE GRAPHS OF REGULAR GRAPHS Kragujevac Journal of Mathematics Volume 39(1) (015), Pages 7 1. SEIDEL ENERGY OF ITERATED LINE GRAPHS OF REGULAR GRAPHS HARISHCHANDRA S. RAMANE 1, IVAN GUTMAN, AND MAHADEVAPPA M. GUNDLOOR 3 Abstract.

More information