let g be a (p,q) graph. an injective map f : e(g) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: v (g) → z - {0} defined by f*(v) = σp∈ev f (e) is one-one where ev denotes the set of edges in g that are incident with a vertex v and f*(v (g)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} u {±k(p+1)/2} according a...

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defined by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or o...

Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

Abstract. In this paper Reverse Wiener index, Reverse Detour Wiener index, Reverse Circular Wiener index Reverse Harary index, Reverse Detour Harary index, Reverse Circular Harary index, Reverse Reciprocal Wiener index, Reverse Detour Reciprocal Wiener index, Reverse Circular Reciprocal Wiener index, Reverse Hyper Wiener index, Reverse, Detour Hyper Wiener index, Reverse Circular Hyper Wiener i...

Let G be a connected graph with vertex set V and a weight function ρ that assigns a nonnegative number to each of its vertices. Then, the ρ-moment of G at vertex u is defined to be M G(u) = ∑ v∈V ρ(v) dist(u, v), where dist(·, ·) stands for the distance function. Adding up all these numbers, we obtain the ρ-moment of G: M G = ∑ u∈V M G(u) = 1 2 ∑ u,v∈V dist(u, v)[ρ(u) + ρ(v)]. This parameter ge...

The edge Szeged index of a connected graph G is defined as the sum of products mu(e|G)mv(e|G) over all edges e = uv of G, where mu(e|G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and mv(e|G) is the number of edges whose distance to vertex v is smaller than the distance to vertex u. In this paper, we determine the n-vertex unicyclic graphs with th...

Let G and H be two graphs. The corona product G o H is obtained by taking one copy of G and |V(G)| copies of H; and by joining each vertex of the i-th copy of H to the i-th vertex of G, i = 1, 2, …, |V(G)|. In this paper, we compute PI and hyper–Wiener indices of the corona product of graphs.

A method for the calculation of the hyper–Wiener index (WW ) of a benzenoid system B is described, based on its elementary cuts. A pair of elementary cuts partitions the vertices of B into four fragments, possessing nrs , r, s = 1, 2 vertices. WW is equal to the sum of terms of the form n11 n22 + n12 n21 . The applicability of the method is illustrated by deducing a general expression for WW of...

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374–385; Adv. Appl. Math. 34 (2005), 138–155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80(1997), 37–56] determined extremal values of σT (w)/σT (u), σT (w)/σT (v), σ(T )/σT (v), and σ(T )/σT (w), where T is ...