EXACT FORMULAE OF GENERAL SUM-CONNECTIVITY INDEX FOR SOME GRAPH OPERATIONS. Shehnaz Akhter, Rashid Farooq and Shariefuddin Pirzada

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1 MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, , September 2018 research paper originalni nauqni rad EXACT FORMULAE OF GENERAL SUM-CONNECTIVITY INDEX FOR SOME GRAP OPERATIONS Shehnaz Akhter, Rashid Farooq and Shariefuddin Pirzada Abstract. Let G be a graph with vertex set V G and edge set EG. The degree of a vertex a V G is denoted by d Ga. The general sum-connectivity index of G is defined as χ αg ab EG dga dgbα, where α is a real number. In this paper, we compute exact formulae for general sum-connectivity index of several graph operations. These operations include tensor product, union of graphs, splices and links of graphs and ajós construction of graphs. Moreover, we also compute exact formulae for general sum-connectivity index of some graph operations for positive integral values of α. These operations include cartesian product, strong product, composition, join, disjunction and symmetric difference of graphs. 1. Introduction and preliminary results Let G V G, EG be a simple and connected graph. An edge with end vertices a and b is denoted by ab or ba. The order and size of graph G are denoted by n G and m G, respectively. The degree of a vertex a V G, denoted by d G a, is the number of vertices incident with a. A path P n of length n 1 is a graph with vertex set {a i i 1,..., n} and edge set {a i a i1 i 1,..., n 1}. A cycle C n of length n is a graph with vertex set {a i i 1,..., n} and edge set {a i a i1 i 1,..., n 1} {a n a 1 }. A complete graph of order n is denoted by K n. A topological index is a mathematical measure which correlate to the chemical structures of any simple finite graph. They play an important role in the study of QSAR/QSPR. There are numerous topological descriptors that have some applications in theoretical chemistry. Among these topological descriptors, the degree based topological indices are of great importance. The first degree based topological indices that are defined by Gutman and Trinajstić [8] in 1972, are the first and second Zagreb indices. These indices are defined as follows: M 1 G d G a 2, M 2 G d G ad G b. ere M 1 G and M 2 G a V G ab EG 2010 Mathematics Subject Classification: 05C07 Keywords and phrases: General sum-connectivity index; graph operations. 267

2 268 Exact formulae of general sum-connectivity index denote the first and second Zagreb index, respectively. Li and Zhao [13] introduced the first general Zagreb index: M α G d G a α, α R. a V G The general Randić index product-connectivity index, introduced by Li and Gutman [12], is defined in the following way: R α G d G ad G b α, α R. ab EG Then R 1/2 G is called the Randić index which was defined by Randić [15] in The general sum-connectivity index is introduced by Zhou and Trinajstić [18] and is defined as: χ α G d G a d G b α, α R. ab EG Then χ 1/2 G is the classical sum-connectivity index which was defined by Zhou and Trinajstić [17] in Another variant of the Randić index of G is the harmonic index, denoted by G and is defined as follows: G 2 d G a d G b 2χ 1G. ab EG Recently Ashrafi et al. [1] computed the exact formulae for Zagreb coindices of some graph operations. Ashrafi et al. [2] calculated some topological indices of splices and links of graphs. In [16], Yarahmadi computed some topological indices of tensor product of graphs. For a detailed study on topological indices of graph operations, we refer the reader to [3, 4, 6, 7, 9 11, 14]. This paper is organized as follows. In Section 2, we present some known graph operations. In Section 3, we give exact formulae of the general sum-connectivity index for several graph operations. These graph operations include tensor product, union of graphs, splices and links of graphs and ajós construction of graphs for real values of α. In Section 4, we compute exact formulae of the general sum-connectivity index of some graph operations including cartesian product, strong product, composition, join, disjunction and symmetric difference of graphs for positive integral values of α. 2. Graph operations Let G and be two simple connected graphs whose vertex sets are disjoint. The set of real numbers and the set of positive integers are denoted by R and Z, respectively. The tensor product of G and, denoted by G, is the graph with vertex set V G V G V and a, bc, d EG whenever ac EG and bd E. The order and the size of G are n G n and 2m G m, respectively. The degree of a vertex a, b V G is given by d G a, b d G ad b. 1

3 S. Akhter, R. Farooq, S. Pirzada 269 The union of G and, denoted by G, is the graph with vertex set V G V G V and edge set EG EG E. The order and size of G are n G n and m G m, respectively. For a V G, either a V G or a V but not both. The degree of a in G is given by { dg a if a V G, d G a d a if a V. Splices of graphs also known as coalescences of graphs were introduced by Doslić [5] in A splice of G and for given vertices a V G and b V, denoted by G a, b, is defined by identifying the vertices a and b in the union of G and. The order and the size of G a, b are n G n 1 and m G m, respectively. Let x be the vertex obtained by identifying a V G and b V. Then V G a, b {x} V G V and EG a, b EG E, respectively. The degree of a vertex c V G a, b is given by d G a,b c d G c if c V G, c a, d c if c V, c b, d G a d b if c x. The motivation for considering these graphs comes from chemistry, where splices of cycles serves as models of spirane molecules and models of complex molecules are built from simpler building blocks by iterating and/or combining the splice and link operations. Links of graphs were introduced by Doslić [5] in The link of G and for given vertices a V G and b V, denoted by G a, b, is obtained by joining the vertices a and b by an edge in the union of G and. The order and the size of G a, b are n G n and m G m 1, respectively. The set of vertices and edges are V G a, b V G V and EG a, b EG V {ab}, respectively. The degree of a vertex c V G a, b is given by d G a,b c d G c if c V G, c a, d c if c V, c b, d G c 1 if c a, d c 1 if c b. Let aá be an edge in G and b b be an edge in. Then the ajós construction of graphs, denoted by G aá, b b, is obtained by identifying a and b, deleting the edges aá and b b, and adding an edge á b. The order and size of G aá, b b are n G n 1 and m G m 1, respectively. Let x be the vertex obtained by identifying a V G and b V. Then V G aá, b b {x} V G\{a} V \{b} and EG aá, b b EG \ {aá} V \ {b b} {á b}. The degree of a vertex c V G aá, b b is given by d G aá,b b c d G c if c V G, c a, d c if c V, c b, d G a d b 2 if c x. The cartesian product of G and, denoted by G, is the graph with vertex set 2 3 4

4 270 Exact formulae of general sum-connectivity index V G V G V and a, bc, d EG whenever [a c and bd E] or [ac EG and b d]. The order and size of G are n G n and m G n m n G, respectively. The degree of a vertex a, b V G is given by d G a, b d G a d b. 5 The strong product of G and, denoted by G, is the graph with vertex set V G V G V and a, bc, d EG whenever [a c and bd E] or [ac EG and b d] or [ac EG and bd E]. The order and size of G are n G n and n G m n m G 2m G m, respectively. The degree of a vertex a, b V G is given by d G a, b d G a d b d G ad b. 6 The composition lexicographic product of G and, denoted by G[], is the graph with vertex set V G[] V G V and a, bc, d EG[] whenever [a is adjacent to c in G] or [a c and b is adjacent to d in ]. The order and size of G[] are n G n and m G n 2 n Gm, respectively. The degree of a vertex a, b V G[] is given by d G[] a, b n d G a d b. 7 The join of G and, denoted by G, is the graph union G together with all the edges joining V G and V. The order and size of G are n G n and m G m n G n, respectively. The degree of a vertex a in G is given by { dg a n d G a if a V G, 8 d a n G if a V. The disjunction of G and, denoted by G, is the graph with vertex set V G V G V and a, bc, d EG whenever ac EG or bd E. The order and size of G are n G n and m G n 2 m n 2 G 2m Gm, respectively. The degree of a vertex a, b in G is given by d G a, b n d G a n G d b d G ad b. 9 The symmetric difference of G and, denoted by G, is the graph with vertex set V G V G V and a, bc, d EG whenever ac EG or bd E but not both. The order and size of G are n G n and m G n 2 m n 2 G 4m Gm, respectively. The degree of a vertex a, b in G is given by d G a, b n d G a n G d b 2d G ad b. 3. Formulae of general sum-connectivity index when α R In this section, we derive exact formulae of general sum-connectivity index for some graph operations defined in Section 2. In the following theorem, we compute the general sum-connectivity index of G. Theorem 3.1. Let G and be two graphs such that either G or is regular. Then

5 S. Akhter, R. Farooq, S. Pirzada 271 the general sum-connectivity index of G is given by the formula: χ α G 1 2 α 1 χ αgχ α. Proof. By equation 1 and the definition of general sum-connectivity index, we have χ α G 2 d G ad b d G cd d α. 10 ac EG bd E Without loss of generality, assume that G is a regular graph. For each a, bc, d EG, we have d G a d G b. Thus d G ad bd G cd d 1 2 d Gad b d G ad b d G cd d d G cd d 1 2 d Gad b d G cd b d G ad d d G cd d d Ga d G cd b d d. Then 10 together with 11 gives χ α G 2 d G ad b d G cd d α 2 ac EG bd E 2 2 α ac EG ac EG bd E 1 2 d Ga d G cd b d d d G a d G c α bd E α d b d d α 1 2 α 1 χ αgχ α. Example 3.2. Applying Theorem 3.1, the general sum-connectivity index for tensor product of some graphs is given below: 1 χ α C n C m 2 3α1 mn, 2 χ α K n K m 1 2 α1 mn[n 1m 1]α. In the following theorem, we give the general sum-connectivity index of union of finite number of graphs. The proof is omitted since it can be easily derived. Theorem 3.3. Let G 1, G 2,..., G n be vertex-disjoint graphs. Then the general sumconnectivity index of G i is given by the following k formula: i1 k χ α G i χ α G 1 χ α G 2... χ α G n. i1 Example 3.4. Using Theorem 3.3, the general sum-connectivity index for union of some graphs is given below: k 1 χ α C ni 4 α k k n i, 2 χ α P ni 2k3 α 4 α 4 α k n i. i1 i1 In the following theorem, we compute the general sum-connectivity index of splices of two graphs G and for given vertices a and b. i1 i1

6 272 Exact formulae of general sum-connectivity index Theorem 3.5. Let G and be two graphs. Then the general sum-connectivity index of G a, b is given by the formula: χ α G a, b χ α G [d G a d b d G d α d G a d G d α ] d N G a χ α [d b d G a d d α d b d d α ]. d N b Proof. By equation 2 and the definition of general sum-connectivity index, we have χ α G a, b d G c d G d α d c d d α cd EG,ca d V G cd EG c,d a d G c d G d d b α cd E c,d b cd E,cb d V d G a d c d d α χ α G d G a d G d α χ α d b d d α d N G a d N b d G a d b d G d α d b d G a d d α d N G a d N b χ α G [d G a d b d G d α d G a d G d α ] d N G a χ α [d b d G a d d α d b d d α ]. d N b Example 3.6. Applying Theorem 3.5, the general sum-connectivity index for splices of K n and C m is given below: χ αk n C ma, b 2 α 1 nn 1 α 4 α m 2 2 α n 1n α n 1 α 2n 3 α. In the following theorem, we compute the general sum-connectivity index of links of two graphs G and for given vertices a and b. Theorem 3.7. Let G and be two graphs. Then the general sum-connectivity index of G a, b is given by the formula: χ α G a, b χ α G χ α [d G a 1 d G d α d G a d G d α ] d N G a d G a d b 2 α [d b 1 d d α d b d d α ]. d N b Proof. Equation 3 and the definition of general sum-connectivity index give χ α G a, b d Gc d Gd α d c d d α cd EG c,d a cd E c,d b

7 S. Akhter, R. Farooq, S. Pirzada 273 d Ga d b 2 α cd EG,ca d V G d Gc 1 d Gd α cd E,cb d V χ αg d Ga d Gd α χ α d b d d α d N G a d N b d c 1 d d α d Ga d b 2 α d Ga 1 d Gd α d b 1 d d α d N G a d N b χ αg χ α [d Ga 1 d Gd α d Ga d Gd α ] d N G a d Ga d b 2 α [d b 1 d d α d b d d α ]. d N b Example 3.8. Applying Theorem 3.7, the general sum-connectivity index of P n C m a, b is given by χ α P n C m a, b 2 α1 2 3 α m mn 34 α. Theorem 3.9. Let G and be two graphs. Then the general sum-connectivity index of G aá, b b is given by the formula: χ α G aá, b b χ α G χ α d G á d b α d G a d b 2 d G d α d G a d G d α á d Na b d Nb d b d G a 2 d d α d N G a d N b d b d d α. Proof. By equation 4 and the definition of general sum-connectivity index, we get χ αg aá, b b d Gc d Gd α d c d d α d Gá d b α cd EG,ca á d V G cd EG c,d a cd E c,d b d Gc d b 2 d Gd α cd E,cb b d V d c d Ga 2 d d α χ αg d Ga d Gá α d Ga d Gd α χ α d b d b α á d N G a d b d d α d Ga d b 2 d Gd α b d N b á d N G a d b d Ga 2 d d α d Gá d b α b d N b χ αg χ α d Gá d b α d Ga d b 2 d Gd α d Na á d N G a d Ga d Gd α d b d Ga 2 d d α d b d d α. b d N b d Nb

8 274 Exact formulae of general sum-connectivity index Example Applying Theorem 3.9, the general sum-connectivity index of C n K m aa, bb is given by: χ α C n K m aa, bb n 24 α 2m 1 α 2 α 1 m 1 α m 2 m Formulae of general sum-connectivity index when α Z In this section, we derive exact formulae of general sum-connectivity index for some graph operations defined in Section 2 when α Z. Theorem 4.1. Let G and be two graphs and α Z. Then the general sumconnectivity index of G is given by the formula: α α χ α G 2 n M n Gχ α n M n χ α n G. Proof. By equation 5 and the definition of general sum-connectivity index, we have χ α G 2d G u d b d d α u V G bd E ac EG v V 2d v d G a d G c α. 12 Now, for u, a, c V G and v, b, d V, using binomial expansion, we obtain α α 2d G u d b d d α 2 n d G u n d b d d α n, 13 α α 2d v d G a d G c α 2 n d v n d G a d G c α n. 14 Using equations 13 and 14 in equation 12, we get χ α G α α 2 n d G u n d b d d α n u V G bd E α ac EG v V α 2 n α α u V G α 2 n 2 n α v V d G u n d v n d v n d G a d G c α n bd E ac EG α α 2 n M n Gχ α n d b d d α n α d G a d G c α n α 2 n M n χ α n G.

9 S. Akhter, R. Farooq, S. Pirzada 275 Example 4.2. Using Theorem 4.1, the general sum-connectivity index of cartesian product of C l and C m is χ α C l C m 2 3α1 lm. In the following theorem, we compute the general sum-connectivity index of G. Theorem 4.3. Let G and be two graphs such that either G or is regular and α Z. Then the general sum-connectivity index of G is given by the formula: α 1 α n χ αg χ ng 2 n β n χ 2 n 1 α nβ 15 α α n M ng 2 n β n n χ αβ n M n 2 n β n χ αβ n G. Proof. By equation 6 and the definition of general sum-connectivity index, we get χ α G 2d G u d b d d d G ud b d d α ac EG v V 2 ac EG bd E u V G bd E 2d v d G a d G c d vd G a d G c α d G a d G c d b d d d G ad b d G cd d α Now, for u V G and b, d V, using binomial expansion, we obtain 2d G u d b d d d G ud b d d α α α n n d G u n 2 n β d b d d α nβ. 16 Similarly, for v V and a, c V G, we obtain 2d v d G a d G c d vd G a d G c α α α n n d v n 2 n β d G a d G c α nβ. 17 Without loss of generality, assume that G is a regular graph. Then for a, bc, d EG, we obtain d Gad b d Gcd d 1 dgadb dgadb dgcdd dgcdd 2 Using binomial expansion, we get 1 dgadb dgcdb dgadd dgcdd 2 1 dga dgcdb dd. 2 d Ga d Gc d b d d d Gad b d Gcd d α

10 276 Exact formulae of general sum-connectivity index α n 1 α d Ga d Gc n 2 n β n d b d d α nβ n 1 Using equations in equation 3, we get χ αg α n α d Gu n 2 n β n d b d d α nβ u V G bd E α n α d v n 2 n β n d Ga d Gc α nβ ac EG v V α n 1 α d Ga d Gc n 2 n β n d b d d α nβ 2 n 1 ac EG bd E α α n d Gu n 2 n β n d b d d α nβ u V G bd E α α n d v n 2 n β n d Ga d Gc α nβ v V ac EG α 1 α n d Ga d Gc n 2 n β n d b d d α nβ. 2 n 1 Thus, 15 holds. ac EG bd E Example 4.4. Using Theorem 4.3, the general sum-connectivity index of strong product of C l and P m is given by: α α n [ n 1 χ α C l P m 2 n β 2 3 αβ 3 m 2n2 l 3 n 1 4 αβ m 2 4 m l ] 1 2 n1 4 β 2 2n 1 m 2. In the following theorem, we give the general sum-connectivity index of composition of two graphs. Theorem 4.5. Let G and be two graphs such that either G or is regular and α Z. Then the general sum-connectivity index of G[] is given by the formula: α α χ α G[] 2 k n k M k Gχ α k α α k k χ α kg M β M k β. 19 k aβ Proof. Equation 7 and the definition of general sum-connectivity index give χ α G[] 2n d G w d u d v α w V G uv E

11 S. Akhter, R. Farooq, S. Pirzada 277 ac EG u V v V n d G a d G c d u d v α. 20 Now, for w V G and u, v V, by using binomial expansion, we get 2n d G w d u d v α α α 2 k n k d G w k d u d v α k. 21 Similarly, for a, c V G and u, v V, we obtain n d G a d G c d u d v α α α k k d Ga d G c α k d u β d v k β. 22 k β Equation 20 along with equations 21 and 22 gives χ αg[] w V G uv E ac EG u V v V α α 2 k n k α α α α 2 k n k d Gw k d u d v α k α α k d Ga d Gc α k k d u a d v k β k β w V G ac EG d Gw k uv E d u d v α k k d Ga d Gc α k k k β u V d u β v V d v k β. Thus the required result is given by 19. Example 4.6. Using Theorem 4.5, we obtain the general sum-connectivity index of the fence graph P n [P 2 ] below: χ αp n[p 2] α α k 2 α k 2 3 α k n 24 α k k [ k β 4 β 4 k1]. k αβ In the following theorem, we compute the general sum-connectivity index of join of finite number of graphs for α Z. Theorem 4.7. Let α Z and G 1, G 2,..., G n be vertex-disjoint graphs with V i V G i and E i EG i, 1 i n, G G 1 G 2... G n and V V G. Then the general sum-connectivity index of join of graphs is given by formula: χ α G n α i1 2 k α V V i k χ α k G i 23

12 278 Exact formulae of general sum-connectivity index 1 2 n α i j,i,j1 α 2 V V i V j α k k k M β G i M k β G j. k β Proof. By equation 8 and the definition of general sum-connectivity index, we obtain n χ α G d Gi u d Gi v 2 V V i α i1 uv E i 1 n d Gi u d Gj v 2 V V i V j α 24 2 v V j i j,i,j1 u V i Now, for u, v V i, 1 i n, we use binomial expansion to obtain d Gi ud Gi v 2 V V i α α α 2 k V V i k d Gi u d Gi v α k. 25 Similarly, for u V G i and v V G j, 1 i, j n, we obtain d Gi ud Gj v 2 V V i V j α α α 2 V V i V j α k k Using equations 25 and 26 in equation 24, we get χ αg n α 2 k α V V i k d Gi u d Gi v α k n α α k 2 V V i V j α k i1 uv E i k d Gi u β d Gj v k β. 26 k β k k β 1 d Gi u β d Gj v k β 2 i j,i,j1 u V i v V j n α 2 k α V V i k d Gi u d Gi v α k i1 uv E i 1 n α α k 2 V V i V j α k k d Gi u β d Gj v k β. 2 k β u V i v V j i j,i,j1 After simplification, we get exactly 23. Example 4.8. Using Theorem 4.7, the general sum-connectivity index of join of P n and P m is given below: α [ α χ αp n P m 2 3 α m k n k 4 α m k n 2 n k m 2 1 k ] 2 n k mα k 4 m 22 2k 2β1 2n 24 β n 2m 24 k. k β

13 S. Akhter, R. Farooq, S. Pirzada 279 In the following theorem, we compute the general sum-connectivity index of G. Theorem 4.9. Let G and be two graphs such that either G or is regular and α Z. Then the general sum-connectivity index of G is given by the formula: χ αg α 1 α 2 k α 1 α 2 k α 1 α 4 2 k G G k k k M ngm k n G n n 1 1 n 2 k n k χ αn k k k k M nm k n n k n 1 n 2 k n k χ αn k G k χ k G 1 n 2 k n n k n k χ α kn. Proof. Equation 9 and the definition of general sum-connectivity index imply χ αg 27 n d Ga d Gc n Gd b d d d Gad b d Gcd d α a V G c V G bd E ac EG b V d V 4 ac EG bd E n d Ga d Gc n Gd b d d d Gad b d Gcd d α n d Ga d Gc n Gd b d d d Gad b d Gcd d α. Now, for a, c V G and b, d V, we obtain the following by using binomial expansion: n d G a d G c n G d b d d d G ad b d G cd d α α 1 α 2 k G k k d G a n d G c k n k k n n 1 1n 2 k n d b d d αn k. 28 Similarly, for a, c V G and b, d V, using binomial expansion, we obtain n G d b d d n d G a d G c d G ad b d G cd d α α 1 α k k 2 k d b n d d k n k k n k n 1n 2 k n d G a d G c αn k 29

14 280 Exact formulae of general sum-connectivity index Without loss of generality, assume that G is a regular graph. EG, we obtain Then a, bc, d d Gad b d Gcd d 1 dgadb dgadb dgcdd dgcdd dgadb dgcdb dgadd dgcdd 1 dga dgcdb dd. 2 Using binomial expansion, we obtain n d G a d G c n G d b d d d G ad b d G cd d α 30 α 1 α k k 2 k G d Ga d G c k 1 n 2 k n n k n d b d d α kn. a V G c V G bd E Using equations in equation 27, we get χ αg α 1 α 2 k k k d Ga n d Gc k n α ac EG b V d V 2 k 1 2 k k n n 1 α G 1 n 2 k n k d b d d αn k k k k d b n d d k n n k n 1 n 2 k n k d Ga d Gc αn k α 1 α k 4 2 k G χ k G 1 n 2 k n n k n k χ α kn ac EG bd E α 1 α k k 2 k G d Ga n d Gc k n a V G c V G k n n 1 1 n 2 k n k d b d d αn k bd E α 1 α k k 2 k d b n d d k n b V d V k n k n 1 n 2 k n k d Ga d Gc αn k ac EG α 1 α k 4 G d Ga d Gc k 1 n 2 k n n k n k bd E ac EG d b d d α kn. Thus the statement of the theorem is true.

15 S. Akhter, R. Farooq, S. Pirzada 281 Example Using Theorem 4.9, the general sum-connectivity index of C l C m is given below: α [ α k k χ αc l C m l α k1 k k m n1 1 n 2 n2α k k m α k2 k k 1 n 2 n2α 4ml α k1 k k ] 1 n 2 n2α. In the following theorem, we give the general sum-connectivity index of symmetric difference of two graphs for α Z. The proof is similar to the proof of Theorem 4.9, hence omitted. Theorem Let G and be two graphs such that either G or is regular and α Z. Then the general sum-connectivity index of G is given by the formula: χ αg α α α α α α 2 G G k k k M ngm k n G n k n 1 n k χ αn k k k k M nm k n n k n G 1 n k χ αn k G k χ k G 1 n n k n k χ α kn. Acknowledgement. The first and second authors are thankful to the igher Education Commission of Pakistan for supporting this research under the grant No /NRPU /R&D/EC/12. The research of the third author is supported by SERB-DST, New Delhi with the research project No. EMR/2015/ References [1] A. R. Ashrafi, T. Doslic, A. amzeh, The Zagreb coindices of graph operations, Discrete. Appl. Math., , [2] A. R. Ashrafi, A. amzeh, S. ossein-zadeh, Calculation of some topological indices of splices and links of graphs, J. Appl. Math. and Informatics, , [3] M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations, Appl. Math. Comput., , [4] N. De, S. M. A. Nayeem, A. Pal, F -index of some graph oprations, arxiv: v1, [5] T. Doslić, Splices, links and their degree weighted Wiener polynomials, Graph Theory Notes N. Y., , [6] M. Eliasi, G. Raeisi, B. Taeri, Wiener index of some graph operations, Discrete Appl. Math., , [7] B. Eskender, E. Vumar, Eccentric connectivity index and eccentric distance sum of some graph operations, Trans. Comb., , [8] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals, Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., , [9] G. ajós, Über eine Konstruktion nicht n-färbbarer Graphen, Wissenschaftliche Zeitschrift d. Wiss. Z. Martin-Luther-Univ. alle-wittenberg, Math.-Naturw. Reihe, ,

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