Recently, it was conjectured by Gutman and Ashrafi that the complete graph Kn has the greatest edge-Szeged index among simple graphs with n vertices. This conjecture turned out to be false, but led Vukičević to conjecture the coefficient 1/15552 of n6 for the approximate value of the greatest edge-Szeged index. We provide counterexamples to this conjecture.

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|...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...

The Szeged index of a graph G, denoted by S z(G) = ∑ uv=e∈E(G) nu (e)n G v (e). Similarly, the Weighted Szeged index of a graph G, denoted by S zw(G) = ∑ uv=e∈E(G) ( dG(u)+ dG(v) ) nu (e)n G v (e), where dG(u) is the degree of the vertex u in G. In this paper, the exact formulae for the weighted Szeged indices of generalized hierarchical product and Cartesian product of two graphs are obtained.

We prove a conjecture of Nadjafi-Arani et al. on the difference between the Szeged and the Wiener index of a graph (M. J. Nadjafi-Arani, H. Khodashenas, A. R. Ashrafi: Graphs whose Szeged and Wiener numbers differ by 4 and 5, Math. Comput. Modelling 55 (2012), 1644–1648). Namely, if G is a 2-connected non-complete graph on n vertices, then Sz (G) −W (G) ≥ 2n − 6. Furthermore, the equality is ob...

The Szeged index of a graph G is defined as S z(G) = ∑ uv = e ∈ E(G) nu(e)nv(e), where nu(e) is number of vertices of G whose distance to the vertex u is less than the distance to the vertex v in G. Similarly, the revised Szeged index of G is defined as S z∗(G) = ∑ uv = e ∈ E(G) ( nu(e) + nG(e) 2 ) ( nv(e) + nG(e) 2 ) , where nG(e) is the number of equidistant vertices of e in G. In this paper,...

The vertex Padmakar-Ivan (PIv) index of a graph G was introduced as the sum over all edges e = uv of G of the number of vertices which are not equidistant to the vertices u and v. In this paper we provide an analogue to the results of T. Mansour and M. Schork [The PI index of bridge and chain graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) 723-734]. Two efficient formulas for calculating th...

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...

Recently the vertex Padmakar–Ivan (PI v) index of a graph G was introduced as the sum over all edges e = uv of G of the number of vertices which are not equidistant to the vertices u and v. In this paper the vertex PI index and Szeged index of bridge graphs are determined. Using these formulas, the vertex PI indices and Szeged indices of several graphs are computed.

the edge szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph g, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. in this paper, the edge szeged index of one-pentagonal carbon nanocone cnc5[n] is computed for the first ...