Some Bounds on General Sum Connectivity Index

Some Bounds on General Sum Connectivity Index Yun Gao Department of Editorial, Yunnan Normal University, Kunming 65009, China, gy6gy@sinacom Tianwei Xu, Li Liang, Wei Gao School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China Abstract In this paper, we determine the lower bound for the general sum connectivity index of molecular graphs with ( G) The extremal molecular structure to reach the lower bound is also presented Furthermore, we consider the lower bound and extremal molecular graph for triangle-free chemical structures Keywords theoretical chemistry, molecule graph, general sum connectivity index, triangle-free IINTRODUCTION In theoretical chemistry, drugs and chemical compounds are modeled as graphs where each vertex represents an atom of molecule and covalent bounds between atoms are expressed by edges between the corresponding vertices The graph obtained from a chemical compounds is often called its molecular graph and can be different structures Chemical indices are introduced to reflect certain structural features of organic molecules Specifically, let G be the class of connected molecular graphs a topological index can be regarded as a score function f: G, with this property that f(g ) = f(g ) if G and G are isomorphic There are several vertex distance-based and degree-based indices which introduced to analyze the chemical properties of molecule graph For instance: Wiener index, PI index, Szeged index, geometric-arithmetic index, atom-bond connectivity index and general sum connectivity index are introduced to test the performance of chemical molecular structures Several papers contributed to determine these distance-based or degree-based indices of special molecular graph (See Yan et al, [], Gao et al, [], Gao and Shi [], Gao and Wang [], Xi and Gao [5-6], Xi et al, [7], and Gao et al, [8] for more detail for more detail) The molecular graphs considered in our paper are all simple The notations and terminologies used but undefined in this paper can be found in Bondy and Mutry [9] The sum connectivity index ( ( G) ) of molecular graph G is defined as: ( G) = uve ( G) ( d( u) d( v)) Few years ago, Zhou and Trinajstic [0] introduced the general sum connectivity as, ( G ) = ( d( u) d( v)) uve ( G) where is a real number Du et al, [] reported the maximum value for the general sum connectivity indices of trees with fixed number of vertex, and the corresponding extremal trees for several special real number are determined Ma and Deng [] computed the tight lower bounds of the sum connectivity index of cacti Xing et al, [] calculated the lower and upper bounds for the sum connectivity indices of tree structure with given numbers of vertices and pendant vertices Du et al, [] yielded the minimum sum connectivity indices of trees and unicyclic graphs with fixed number of vertices and matching number, respectively, and the corresponding molecular extremal graphs are deduced Furthermore, they obtained the first and second minimum sum connectivity indices of the unicyclic graphs with vertex number at least Du et al, [5] derived the minimum and the second minimum values wwwjmestorg JMESTN5086 6

of the general sum connectivity indices of unicyclic molecular graphs with non-zero and given vertex number Moreover, they provided the corresponding molecular extremal graphs Chen et al, [6] learned the general sum connectivity index of benzenoid systems and phenylenes Du and Zhou [7] studied the sum connectivity index of bicyclic molecular graphs Yang et al, [8] reported the computational formulas for calculating the sum connectivity index of polyomino chains Chen and Li [9] obtained the sharp lower bound of the sum connectivity index for unicyclic molecular graphs with given number of vertex and fixed number of pendent vertices Farahani [0] deduced the sum connectivity index of special classes of nanotubes Very recently, Tache [] obtained the molecular graph with the maximum general sum connectivity index among the connected bicyclic molecular structures with given vertex number and In this paper, our contributions are two-fold We first discuss the tight lower bound general sum connectivity index for molecular graphs with ( G) The sufficient and necessary condition to reach the lower bound is given Then, we focus on the triangle-tree molecular structures The corresponding sharp lower bound and extremal structure are presented in triangle-tree setting IIMINIMUM GENERAL SUM CONNECTIVITY INDEX OF MOLECULAR GRAPH WITH ( G) AND <0 For an edge e=uv of a molecular graph G, its general weight is denoted as ( d( u) d( v)) Lemma Let e be an edge with maximal general weight in G, and <0 We have ( G e ) < ( G) Proof Let e=uv Since edge uv has maximal general weight in molecular graph G, we get dw ( ) dv () for any w Nu ( ) and dw ( ) du ( ) for any w Nv () x Noticing that the function ( x ) is increasing for x > and negative real number, we obtain ( G ) - ( G e) = ( d ( u ) d ( v )) wn ( u)\{ v} (( ( ) ( )) d u d w ( d( u) d( w) ) ) wn ( v)\{ u} (( ( ) ( )) d v d w ( d( v) d( w) ) ) ( d( u) d( v)) ( ( ) )(( ( ) ( )) d u d u d v ( d( u) d( v) ) ) ( ( ) )(( ( ) ( )) d v d u d v ( d( u) d( v) ) ) = ( ( ) ( ) ) d u d v ( d( u) d( v)) >0 Hence, we yield the desired result Let K ab, be the complete bipartite molecular graph with a and b vertices in its two partite sets, respectively For n, the molecular graph, n K is deducedby joining an edge between the two non-adjacent vertices of degree n- in K, n By simple calculation, we infer ( K ) =, n f ( n, ) = ( n ) ( n)( n ) We use ( G) to denote the minimum degree of the molecular graph G Theorem Let G be a molecular graph with vertex number n and minimum degree ( G) Suppose <0 Then ( G ) f (, ) n with equality if and only if G, n K Proof It is not hard to chec that the assertion is hold for n= Suppose it holds for n < n Then, we show that it also holds for n in the following wwwjmestorg JMESTN5086 6

Let G be a molecular graph with at least 5 vertices If ( G) according to Lemma, the deletion of an edge with maximal general weight gets a graph G of minimal degree at least two such that ( G ') < ( G) So, in what follows, we only need to verify the result is hold for molecular graph G with minimum degree Case Each pair of adjacent vertices of degree has a common neighbor Let u and u be a pair of adjacent vertices with degree in molecular graph G, and u is their common neighbor We immediately get du ( n- Subcase If du ( =, let G =G-{ u }, then ( G ) f ( n, ) in view of the induction hypothesis, and f ( n, ) ( du ( ) - ( du ( ) f ( n, ) ( n ) - ( n ) > f ( n, ) Subcase If du ( =, let u be the neighbor of u in G different from u and u, where du ( ) n (i) Suppose that du ( ) = Denote by u 5 the neighbor of u in G different from u, where du ( 5) n Let G = G u uu 5 ( G ) f ( n, ) by the induction hypothesis Note that x ( x ) is decreasing for x > 0 and <0 ( G ) = ( G 5 ( du ( 5) ) - ( du ( 5) ( G ) = ( G ) h ( n, ) > G ( ) 5 ( n ) - ( n ) f ( n, ) 5 f ( n, ) ( n ) - ( n ) > f ( n, ) Subcase If du (, let G = G-{ u }, then ( G ) f ( n, ) in terms of the induction hypothesis Since x ( x ) is increasing for x > and <0, we infer ( G ) = ( G ) vn ( u )\{ u } ( du ( ) (( d( u ) d( v)) ( d( u ) d( v) ) ) ( G ) ( du ( ) ( ( )(( ( ) d u d u ( d( u) ) ( G ) ( du ( ) - ( du ( ) (ii) Suppose that du ( ) n- Let G = G u u u ( G ) f ( n, ) by the induction hypothesis Note that ( x ) ( x ) ( x) is decreasing for x > 0 and <0 ( G ) = ( G ) 5 vn ( u)\{ u} ( du ( ) (( d( u ) d( v)) ( d( u ) d( v) ) ) ( G ) 5 ( du ( ) ( ( ) )(( ( ) ) d u d u ( d( u) ) ) ( G ) wwwjmestorg JMESTN5086 6

5 ( du ( ) -( du ( ) ) ( du ( ) ) ( d( v) d( w)) ( G 6) f ( n, n, ) G ( ) 5 n - ( n ) ( n ) f ( n, ) ( ) n n > f ( n, ) f ( n, ) f ( n, ) 5 n - ( n ) ( n ) > Case There is a pair of adjacent vertices of degree two without common neighbor Let u and u be a pair of adjacent vertices with degree two in G which has no common neighbor Denote by u the neighbor of u in G different from u Let G 5 =G u uu ( G ) f ( n, ) by the induction hypothesis, 5 and ( G ) = ( G 5) f ( n, ) > f ( n, ) Case There is no pair of adjacent vertices of degree two Let u be a vertex of degree two with neighbors v and w in G Subcase vw E, where dv () n and dw ( ) n Let G 6 =G u vw ( G ) f ( n, ) 6 by the induction hypothesis Note that f ( x, y, ) = ( x ) ( y ) ( x y) f ( n, n, ) y n and <0, since for x n, f x < 0 and f y < 0 ( G ) = ( G 6) ( dv ( ) ) ( dw ( ) ) - Subcase vw E, where dv () n- and dw ( ) n- Let G 7 = G-u ( G ) f ( n, ) 7 by the induction hypothesis Note that g( x, y, ) = ( x y) ( x ) ( y ) - ( x y) - ( x ) - ( y ) g( x, y, ) for x n-, g g y n- and <0, since ( ) y x g x g( x,, ) x g y g(, y, ) <0 y g g <0, and ( ) x y <0 and < 0 and ( G ) = ( G 7) ( dv ( ) ) ( dw ( ) ) - ( d( v) d( w) ) (( ( ) ( )) d v d z ( d( v) d( z) ) ) zn ( v)\{ u, w} zn ( w)\{ u, v} (( ( ) ( )) d w d z ( d( w) d( z) ) ) ( ) ( dv ( ) ) ( dw ( ) ) - G 7 ( d( v) d( w) ) ( ( ) )(( ( ) ) d v d v ( d( v) ) ) ( ( ) )(( ( ) ) d w d w ( d( w) ) ) ( ) ( d ( v ) d ( w )) ( dv ( ) ) G 7 wwwjmestorg JMESTN5086 65

( dw ( ) ) - ( d( v) d( w) ) - ( dv ( ) ) - ( dv ( ) ) - ( dw ( ) ) ( G 7) g( n, n, ) f ( n, ) (n ) 6n - (n ) - 6( n ) = f ( n, ) with equality if and only if G K, n Hence, the assertion is true for all n III A LOWER BOUND FOR THE GENERAL SUM CONNECTITY INDEX OF TRIANGLE-FREE MOLECULAR GRAPH WITH ( G) In the section, we will give a best possible lower bound for the general sum connectivity index of a triangle-free molecular graph with ( G) characterize the extremal molecular graphs and Theorem Let G be a triangle-free molecular graph of order n with ( G) Assume <0 Then ( G ) f ( n, ) = ( ) n n with equality if and only if G K, n Proof It is easy to chec that the assertion is true for n= Suppose it holds for n < n; we next show that it also holds for n Let G be a molecular graph with n> vertices If ( G) by Lemma, the deletion of an edge with maximal general weight yields a graph G of minimal degree at least two such that ( G ') < ( G) So, we only need to prove the result is true for G with ( G) = Case There exists a vertex u of degree two such that the neighbors of u have degree at least three Let Nu ( ) ={ u } and du ( i ) n for i = ; ( Gu) and G u is triangle-free ( Gu ) f ( n, ) induction hypothesis by the ( G ) = ( G u) ( du ( ) ) ( du ( ) ) (( ( ) ( )) d u d v ( d( u) d( v) ) ) vn ( u )\{ u} vn ( u )\{ u} (( d( u ) d( v)) ( d( u ) d( v) ) ) ( G u ) ( du ( ) ) ( du ( ) ) ( d( u ) )(( d( u ) ) ( d( u ) ) ) ( d( u ) )(( d( u ) ) ( d( u ) ) ) ( G u ) ( du ( ) ) - ( du ( ) ) ( du ( ) ) - ( du ( ) ) ( G u ) ( n ) - n ( n ) - n f ( n, ) ( n ) - n = f ( n, ) with equality if and only if G K, n Case Every vertex u of degree two has a neighbor of degree two in G Let Nu ( ) ={ u } and du ( ) =, du ( ) ; Nu ( ) ={u,v} Subcase v is not a neighbor of u Let G 8 = G u uu 8 ( G ) and 8 G is triangle-free ( G 8) f ( n, ) by the induction wwwjmestorg JMESTN5086 66

hypothesis ( G ) = ( G 8) f ( n, ) > f ( n, ) Subcase v is also a neighbor of u (I) If dv () = du ( ) =, let G 9 =G u v u u, ( G ) and 9 then 9 n 8 ( G 9) hypothesis f ( n, ) ( G ) = ( G 9) f ( n, ) G is triangle-free, implying by the induction f ( n, ) > (II) If none of v has degree two dv () n and du ( ) n- since G is triangle-free Let G 0 = G u u ( G0) and G 0 is triangle-free, implying n 6 ( G ) f ( n, ) by the induction hypothesis 0 (( ( ) ( )) d w d v ( d( w) d( v) ) ) wn ( v)\{ u} wn ( u )\{ u, v} (( d( u ) d( w)) ( d( u ) d( w) ) ) ( G 0) ( dv ( ) ) ( du ( ) ) ( d( v) d( u )) - ( d( v) d( u) ) ( d( v) )(( d( v) ) ( d( v) ) ) ( d( u ) )(( d( u ) ) ( d( u ) ) ) G 0 ( ) t( d( v), d( u), ) G 0 ( ) t( n, n, ) f ( n, ) t( n, n, ) > f ( n, ) (III) If exactly one of v has degree two, without loss of generality, assume du ( ) = dv () n since G is triangle-free Note that (i) If dv (), let G = G u u u t( x, y, ) = ( x y) - ( x y ) ( x ) ( y ) - ( x ) - ( y ) t( n, n, ) t for x n, y n and <0, since ( ) <0 y x and t x t( x,, ) <0 and x t y <0, similarly ( G ) = ( G 0) ( dv ( ) ) ( du ( ) ) ( d( v) d( u )) - ( d( v) d( u) ) ( G ) and G is triangle-free, implying n 7 ( G ) f ( n, ) by the induction hypothesis ( G ) = ( G ) wn ( v)\{ u} ( dv ( ) ) (( ( ) ( )) d w d v ( d( w) d( v) ) ) ( G ) ( dv ( ) ) ( ( ) )(( ( ) ) d v d v ( d( v)) ) ( ) ( dv ( )) - ( dv ( ) ) G wwwjmestorg JMESTN5086 67

G ( ) ( n - ( n ) ( d( u ) )(( d( u ) ) ( d( u ) ) ) f ( n, ) ( n - ( n ) > f ( n, ) (ii)if d(v) =, denote by u the neighbor of v in G different from u and u (a) If du ( =, let u be the neighbor of u in G different from v and G = G u vu ( G ) and G is triangle-free ( G ) f ( n, ) by the induction hypothesis And G ( ) 5 ( du ( ) - ( d( u ) ) ( d( u ) ) G ( ) 5 ( n ) - ( n ( n ) f ( n, ) 5 ( n ) -( n ( n ) > f ( n, ) The proof of our theorem is completed IVSUM CONNECTIVITY INDEX OF MOLECULAR GRAPH AND TRIANGLE-FREE MOLECULAR STRUCTURE WITH ( G) ( G ) = ( G ) ( G ) 5-5 = 5 ( du ( ) ) - ( du ( ) ( G ) f ( n, ) > f ( n, ) (b) If du ( du ( n 5 as G is triangle-free Let G = G u v u u, we have ( G ) and G is triangle-free, implying n 8 ( G ) f ( n, ) by the induction hypothesis Note that ( x - ( x ) ( x ) is Let =, we get the following results on sum connectivity index of molecular graph Lemma If e is an edge with maximal general weight in G, and <0 ( G e) < ( G) Theorem Let G be a molecular graph with vertex number n and minimum degree ( G) ( G) f ( n, ) = with equality if and only if G Then ( n ) ( n)( n ) K, n Theorem Let G be a triangle-free molecular decreasing for x 0 and <0 graph of order n with ( G) Then ( G ) = ( G wn ( u )\{ v} 5 ( du ( ) (( d( u ) d( w)) ( d( u ) d( w) ) ) G ( ) 5 ( du ( ) ( G) f ( n, ) = ( n) n with equality if and only if G K, n wwwjmestorg JMESTN5086 68

VCONCLUSION In this paper, by virtue of molecular graph structural analysis and mathematical derivation, we determine the lower bound of the general harmonic index of molecular graph with ( G) Furthermore, the lower bound for the general harmonic index of triangle-free molecular graph with ( G) deduced VIACKNOWLEDGEMENTS is We than the reviewers for their constructive comments in improving the quality of this paper This wor was supported in part by NSFC (no 059) We also would lie to than the anonymous referees for providing us with constructive comments and suggestions REFERENCES [] L Yan, Y Li, W Gao, J Li, On the extremal hyper-wiener index of graphs, Journal of Chemical and Pharmaceutical Research, 0, 6(: 77-8 [] W Gao, L Liang, Y Gao, Some results on wiener related index and shultz index of molecular graphs, Energy Education Science and Technology: Part A, 0, (6): 896-8970 [] W Gao, L Shi, Wiener index of gear fan graph and gear wheel graph, Asian Journal of Chemistry, 0, 6(): 97-00 [] W Gao, W F Wang, Second atom-bond connectivity index of special chemical molecular structures, Journal of Chemistry, Volume 0, Article ID 9065, 8 pages, http://dxdoiorg/055/0/9065 [5] W F Xi, W Gao, Geometric-arithmetic index and Zagreb indices of certain special molecular graphs, Journal of Advances in Chemistry, 0, 0(): 5-6 [6] W F Xi, W Gao, -Modified extremal hyper-wiener index of molecular graphs, Journal of Applied Computer Science & Mathematics, 0, 8 (8): -6 [7] W F Xi, W Gao, Y Li, Three indices calculation of certain crown molecular graphs, Journal of Advances in Mathematics, 0, 9(6): 696-0 [8] Y Gao, W Gao, L Liang, Revised Szeged index and revised edge Szeged index of certain special molecular graphs, International Journal of Applied Physics and Mathematics, 0, (6): 7-5 [9] J A Bondy, U S R Mutry Graph Theory, Spring, Berlin, 008 [0] B Zhou, N Trinajstic, On general sum-connectivity index, Journal of Mathematical Chemistry, 00, 7: 0-8 [] Z B Du, B Zhou, N Trinajstic, On the general sum-connectivity index of trees, Applied Mathematics Letters, 0, : 0 05 [] F Y Ma, H Y Deng, On the sum-connectivity index of cacti, Mathematical and Computer Modelling, 0, 5: 97 507 [] R D Xing, B Zhou, N Trinajstic, Sum-connectivity index of molecular trees, Journal of Mathematical Chemistry, 00, 8: 58 59 [] Z B Du, B Zhou, N Trinajstic, Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number, Journal of Mathematical Chemistry, 00, 7:8 855 [5] Z B Du, B Zhou, N Trinajstic, Minimum general sum-connectivity index of unicyclic graphs, Journal of Mathematical Chemistry, 00, 8:697 70 [6] S B Chen, F L Xia, J G Yang, On general sum connectivity index of benzenoid systems and phenylenes, Iranian Journal of Mathematical Chemistry, 00, (): 97-0 [7] Z B Du, B Zhou, On sum-connectivity index of bicyclic graphs, Bull Malays Math Sci Soc, 0, 5():0 7 [8] J G Yang, F L Xia, S B Chen, On sum-connectivity index of polyomino chains, Applied Mathematical Sciences, 0, 5(6): 67 7 wwwjmestorg JMESTN5086 69

[9] J J Chen, S C Li, On the sum-connectivity index of unicyclic graphs with pendent vertices, Math Commun, 0, 6:59-68 [0] M R Farahani, On the randic and sum-connectivity index of nanotubes, Analele Universitatii de Vest, Timisoara Seria Matematica Informatica LI, 0, : 9 6 [] R M Tache, General sum connectivity index with for bicyclic graphs, MATCH Commun Math Comput Chem, 0, 7: 76-77 wwwjmestorg JMESTN5086 70