BoundsonVertexZagrebIndicesofGraphs

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1 Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 6 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. USA) Online ISSN: & Print ISSN: Bounds on Vertex Zagreb Indices of Graphs By K. Pattabiraman & M. Seenivasan Annamalai University Abstract- The vertex version of first and second Zagreb indices were introduced by Tavakoli et al. [] are defined as M G)= d G u)+d G v and M G)= d G u)d G v). In this paper, we obtained the relation between the first and second vertex Zagreb indices of graphs using some well-known results. Keywords: degree, zagreb index, topological index. GJSFR-F Classification: MSC 00: 7B69 BoundsonVertexZagrebIndicesofGraphs {u,v} VG) {u,v} VG) Strictly as per the compliance and regulations of: 07. K. Pattabiraman & M. Seenivasan. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Ref 4. M. Dehmer, K. Varmuza, D. Bonchev, editors 0) Statistical Modelling of Molecular Descriptors in QSAR/QSPR. Quantitative and Network Biology. Wiley- FBlackwell. Bounds on Vertex Zagreb Indices of Graphs Abstract- The vertex version of first and second Zagreb indices were introduced by Tavakoli et al. [] are defined as M G)= d G u)+d G v M and G)= d G u)d G v). In this paper, we obtai ned the relation {u,v} VG) {u,v} VG) between the first and second vertex Zagreb indices of graphs using some well-known results. Keywords: degree, zagreb index, topological index. I. Introduction A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or a matrix which represents the whole graph, and these representations are aimed to be uniquely defined for that graph. A topological index is a numeric quantity associated with a graph which characterize the topology of graph and is invariant under graph automorphism. There are some major classes of topological indices such as distance based topological indices, degree based topological indices and counting related polynomials and indices of graphs. Among these classes degree based topological indices are y and particularly in chemistry. In more precise way, a topological index TopG) of a graph G, is a number with the property that for every graph H isomorphic to G, TopG)=TopH). The topological indices are graph invariants which has been used for examine quantitative structure-property relationships QSPR) and quantitative structureactivity relationships QSAR) extensively in which the biological activity or other properties of molecules are correlated with their chemical structures, see [4]. For a molecular) graph G, The first Zagreb index M G) is the equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M G) is the equal to the sum of the products of the degrees of pairs of adjacent vertices, that is, M G)= dg u)= u VG) d G u)+d G v)), M G) = d G u)d G v), where d G v) is a degree of a vertex v uv EG) uv EG) in G. There are various study of different versions of Zagreb indices. One of the modified versions of classical Zagreb indices, the vertex version of first and second Zagreb indices were introduced by Tavakoli et al. [] are defined as M G)= d G u)+d G v)) and M G)= {u,v} VG) d G u)d G v). {u,v} VG) Another topological index, defined as sum of cubes of degrees of all the vertices was also introduced in the same paper, where the first and second Zagreb indices were introduced [4]. Furtula and Gutman in [3] recently investigated this index and named this index as forgotten topological index or) F -index and showed that the predictive ability of this index is almost similar to that of first Zagreb index and for the entropy and acetic factor, both of them yield correlation coefficients greater than The F -index of a graph G is defined as F=FG)= dg 3 u)= dg u)+d G v)). u VG) uv EG) K. Pattabiraman α & M. Seenivasan σ Global Journal of Science Frontier Research F ) Volume XVII Issue VI V ersion I Year Author α σ: Department of Mathematics, Annamalai University, Annamalainagar , India. pramank@gmail.com 07 Global Journals Inc. US)

3 Global Journal of Science Frontier Research F ) Volume XVII Issue VI V ersion I Year For the survey on theory and application of Zagreb indices, see []. Feng et al.[0] have given a sharp bounds for the Zagreb indices of graphs with a given matching number. Khalifeh et al. [3] have obtained the Zagreb indices of the Cartesian product, composition, join, disjunction and symmetric difference of graphs. Ashrafi et al. [9] determined the extremal values of Zagreb coindices over some special class of graphs. Hua and Zhang [] have given some relations between Zagreb coindices and some other topolodical indices. In [], the vertex version of Zagreb indices for the generalized hierarchical product of two connected graphs are computed. The vertex Zagreb indices of different types of operations of graphs are computed by De []. In this sequence, we have obtained the relation between the first and second vertex Zagreb indices of graphs. In this section, we present the relation between the first and second vertex Zagreb indices of graph. The inverse degree of a graph G is the sum of reciprocal of the vertex degrees, that is, IDG)= u VG) d G u). Lemma.. [5]Let x= x, x,..., x N ) and y= y, y,...,y N ) be sequences of real numbers z= z, z,...,z N ) and w= w, w,...,w N ) be nonnegative sequences, then N w i i= i= N z i xi+ N z i i= i= N N w i y i z i x i i= N w i y i. In particular, if z i and w i are positive, then the equality holds in.) if and only if x= y= k, where k= k, k,...,k) is a constant sequence. Theorem.. Let G be a connected graph with n vertices and m edges. Then M G)M G) n )IDG)+ n )δ δ+ + n )n ) + mm G) FG) 4 with equality if and only if G is regular. Proof. Note that each i in Lemma. corresponds a vertex pair{u i, u j } such that N= nn ). setting z i = w i = x i y i and each x i is replaced by d G u i )d G u j ) and each y i is replaced by, then we obtain d G u i )+d G u j ) II. d G u i )d G u j ) d G u i )+d G u j ) ) Main Results i= d G u i )+d G u j ) d G u i )d G u j ) dg u i )+d G u j ) ) + d Gu i )d G u j ) ) d G u i )+d G u j ) d G u i )d G u j ). One can observe that d G u i ) + d G u j ) δ, it immediately follows that d G u i )d G u j ) d G u i )+d G u j ). Also δ + d G u i ) + δ, we have d Gu i )δ d G u i )+δ δ δ+. Let v k be the minimum degree vertex in G. Then d G u i )+d G u j ) d G u i )d G u j ) + d Gu i )d G u j ) ) d G u i )+d G u j ) = d G u i )+d G u j ) d G u i )d G u j ) = +.).).3) d G u i )d G u j ) d G u i )+d G u j ) d G u i ) + ) d G u i )δ + d G u j ) d {u i,u k } VG) G u i )+δ Ref 0. L. Feng, A. Ili c, Zagreb, Harary and hyper- Wiener indices of graphs with a given matching number, Appl. Math. Lett. 3 00) Global Journals Inc. US)

4 Notes + u j u k n where IDG) is the inverse degree index of G. Since d G u i )=m, then we have d G u i )d G u j ) d G u i )+d G u j ) n d G u i ) + n )δ δ+ + nn ) n ) ) = n )IDG)+ n )δ δ+ + n )n ), 4 d G u i )d G u j ) d G u i )+d G u j ) ) From the definitions of first Zagreb index and F -index of G, we obtain d G u i ) m d G u i ) ). d G u i )d G u j ) d G u i )+d G u j ) ) mm G) FG). Using.4) and.5) in.3), we obtain the required result in.). By Lemma., the equality in.) holds if d G u i )d G u j ) = d G u i )d G u k ) = d G u i )+d G u j )+ d G u i )+d G u k ) for any three vertices u i, u j and u k in G. This implies G is regular. Proposition.3. Let p be the number of pendent vertices in G. Then FG) δm G)+ p δ). Proof. Since p is the number of pendent vertices in G, FG)= d G u i ) 3 = p+ d G u i ) d G u i ) 3 p+δ d G u i ) d G u i ) = p+δm G) p). Using Proposition.3 in Theorem., we obtain the following corollary. Corollary.4. Let G be a connected graph with n vertices and m edges. If p is the number of pendent vertices in G, then M G)M G) n )IDG)+ n )δ δ+ + n )n ) + m δ)m G)+ pδ ) 4 with equality if and only if G is regular. Bounds on Vertex Zagreb Indices of Graphs Lemma.5. Radon s inequality) For real numbers p>0, a.a,...,a N 0 and b, b,..., b N > 0, the following inequality holds: N i= a p+ i b p i N ) p+ a i i= N ). p b i i=.4).5) Global Journal of Science Frontier Research F ) Volume XVII Issue VI V ersion I Year Global Journals Inc. US)

5 Global Journal of Science Frontier Research F ) Volume XVII Issue VI V ersion I Year Next we obtain the another relation between the first and second vertex Zagreb indices of graphs. Theorem.6. Let G be a connected graph with n vertices and m edges. Then ) M G) M G) nn )δ+ ) δ with equality if and only if G is regular. Proof. For each i in Lemma.5 corresponds a vertex u i, u j ) with N= nn ) and p=, setting each a i = d G u i )+d G u i ) and b i = d G u i )d G u i ), it follows that dg u i )+d G u i ) ) d G u i )+d G u i ) ) u i,u j ) VG)..6) d G u i )d G u i ) d G u i )d G u i ) u i,u j ) VG) The equation.6) is equivalent to u i,u j ) VG) It has been proved in [5] that Hence M G) ) M G) d G u i ) d G u j ) + d G u i ) d G u j ) + u i,u j ) VG) d G u j ) d G u i ) δ+ ) Equality in.7) holds if and only if G is regular. δ u i,u j ) VG) d G u j ) d G u i ) ) M G) ) M G) ) δ+ ) δ. = nn )δ+ ). δ References Références Referencias. N. De, The vertex Zagreb indices of some graph operations, Carpathian Math. Publ. 806)5-3.. M. Tavakoli, F. Rahbarnia, A.R. Ashrafi, Eccentric connectivity and Zagreb coindices of the generalized hierarchical product of graphs, J. Discrete Math. doi:0.55/04/ M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math ) M. Dehmer, K. Varmuza, D. Bonchev, editors 0) Statistical Modelling of Molecular Descriptors in QSAR/QSPR. Quantitative and Network Biology. Wiley- FBlackwell. 5. T. Doslic, B. Furtula, A. Graovac, I. Gutman, S. Moradi, Z. Yarahmadi, On vertex degree based molecular structure descriptors, MATCH Commun. Math.Comput. Chem., 66 0) T. Doslic, T. Reti, D. Vukicevic, On the vertex degree indices of connected graphs, Chem. Phys. Lett., 5 0) B. Furtula, I. Gutman, M. Dehmer, On structure sensitivity of degree based topological indices, Appl. Math. Comput., 9 03) ) Notes 07 Global Journals Inc. US)

6 Notes 8. I. Gutman, Degree based topological indices, Croat. Chem. Acta, 86 03) A.R. Ashrafi, T. Doslic, A. Hamzeha, Extremal graphs with respect to the Zagreb coindices, MATCH Commun. Math.Comput. Chem., 65 0) L. Feng, A. Ili c, Zagreb, Harary and hyper- Wiener indices of graphs with a given matching number, Appl. Math. Lett. 3 00) I. Gutman, K. C. Das, The first Zagerb index 30 years after, MATCH Commun. Math. Comput. Chem ) H. Hua, S. Zhang, Relations between Zagreb coindices and some distance-based topological indices, MATCH Commun. Math.Comput. Chem., 68 0) B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem., 534) 05) I. Gutman and N. Trinajsti c, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 7 97) K.C. Das, On geometrical-arithmetic index of graphs, MATCH Commun. Math.Comput. Chem., 64 00) Global Journal of Science Frontier Research F ) Volume XVII Issue VI V ersion I Year Global Journals Inc. US)

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