eccentric connectivity index

Let G be a molecular graph. The eccentric connectivity index, ξ(G) , is defined as, ξ(G)=∑deg(u).ecc(u) , where deg(u) denotes the degree of vertex u and ecc(u) is the largest distance between u and any other vertex v of G. In this paper, an exact formula for the eccentric connectivity index of nanostar dendrimer NS3[n] is given.

Let $G$ be a connected graph on $n$ vertices. $G$ is called tricyclic if it has $n 2$ edges, and tetracyclic if $G$ has exactly $n 3$ edges. Suppose $mathcal{C}_n$ and $mathcal{D}_n$ denote the set of all tricyclic and tetracyclic $n-$vertex graphs, respectively. The aim of this paper is to calculate the minimum and maximum of eccentric connectivity index in $mathcal{C}_n$ and $mathcal{D}_n$.

The eccentric connectivity index of a graph is defined as E(Γ)=∑uεV(Γ)degΓ(u)e(u), where degΓ(u) denotes the degree of the vertex u in Γ and e(u) is the eccentricity of vertex u. In this paper, the modified eccentric connectivity index of two infinite classes of fullerenes is computed.

Let $G$ be a non-abelian group. The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|$. Moreover, we determine eccentric connectivity index of $Gamma_G$ for every non-abelian finite group $G$ in terms of the number of conjugacy classes $k(G)$ and the size of the group $G$.

Let dn;m = 2n+1 and En;m be the graph obtained from a path Pdn;m+1 = v0v1:::vdn;m by joining each vertex of Kn by joining each vertex of Kn. Zhang, Liu and Zhou [On the maximal eccentric connectivity indices of graphs, Appl. Math. J. Chinese Univ., in press] conjectured that if dn;m > 3, then En;m is the graph with maximal eccentric connectivity ndex among all connected graph with n vertices and m edges. In this note, we prove this conjecture. Moreover, we present the graph with maximal eccentric connectivity index among the connected graphs with n vertices. Finally, the minimum of this graph invariant n the classes of tricyclic and tetracyclic graphs are computed.

Let $G=(V,E)$ be a connected graph. The eccentric connectivity index of $G$, $xi^{c}(G)$, is defined as $xi^{c}(G)=sum_{vin V(G)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. The eccentric distance sum of $G$ is defined as $xi^{d}(G)=sum_{vin V(G)}ec(v)D(v)$, where $D(v)=sum_{uin V(G)}d_{G}(u,v)$ and $d_{G}(u,v)$ is the distance between $u$ and $v$ in $G$. In this paper, we calculate the eccentric connectivity index and eccentric distance sum of generalized hierarchical product of graphs. Moreover, we present explicit formulae for the eccentric connectivity index of $F$-sum graphs in terms of some invariants of the factors. As applications, we present exact formulae for the values of the eccentric connectivity index of some graphs of chemical interest such as $C_{4}$ nanotubes, $C_{4}$ nanotoris and hexagonal chains.

The eccentric connectivity index of the molecular graph is defined as $\zeta^c(G)=\sum_{uv\in E}degG(u)ε(u)$ , where degG(x) denotes the degree of the vertex x in G and ε(u)=max{d(x,u) |x ε V(G)}. In this paper this polynomial is computed for an infinite class of fullerenes.

Fullerenes are carbon-cage molecules in which a number of carbon atoms are bonded in a nearly spherical configuration. The connective eccentric index of graph G is defined as C (G)= Σa V(G)deg(a)ε(a) -1, where ε(a) is defined as the length of a maximal path connecting a to another vertex of G. In the present paper we compute some bounds of the connective eccentric index and then we calculate this topological index for two infinite classes of fullerenes.