The vertex-edge Wiener index of a simple connected graph G is defined as the sum of distances between vertices and edges of G. Two possible distances D_1(u,e|G) and D_2(u,e|G) between a vertex u and an edge e of G were considered in the literature and according to them, the corresponding vertex-edge Wiener indices W_{ve_1}(G) and W_{ve_2}(G) were introduced. In this paper, we present exact form...

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 ...

A fuzzy graph is a symmetric binary fuzzy relation on a fuzzy subset. The concept of fuzzy sets and fuzzy relations was introduced by L.A.Zadeh in 1965cite{zl} and further studiedcite{ka}. It was Rosenfeldcite{ra} who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975. The concepts of fuzzy trees, blocks, bridges and cut nodes in fuzzy graph has been studi...

Let γ : E(G) −→ N∗ be an edge colouring of a graph G and σγ : V (G) −→ N∗ the vertex colouring given by σγ(v) = ∑ e3v γ(e) for every v ∈ V (G). A neighbour-sumdistinguishing edge-colouring of G is an edge colouring γ such that for every edge uv in G, σγ(u) 6= σγ(v). The study of neighbour-sum-distinguishing edge-colouring of graphs was initiated by Karoński, Łuczak and Thomason [8]. They conjec...

The local antimagic total vertex labeling of graph G is a that every vertices and edges label by natural number from 1 to such two adjacent has different weights, where sum the labels all incident vertex. If start smallest then edge so kind coloring called super labeling. That induces for v, weight w(v) color v. minimum colors obtained chromatic G, denoted χlsat(G). In this paper, we consider G...

Let G = (V, E) be a graph and k > 2 an integer. The general chromatic index x;(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph (F) induced by F has size at most k1. This paper initiates a study of x;(G) and generalizes some known results on chromatic index. The purpose of this paper is to obtain a generalization of chromatic index....

A k-edge-weighting of G is a mapping ω:E(G)⟶{1,…,k}. The edge-weighting naturally induces vertex-colouring σω:V(G)⟶N given by σω(v)=∑u∈NG(v)ω(vu) for every v∈V(G). ω neighbour sum distinguishing if it yields proper σω, i.e., σω(u)≠σω(v) edge uv G. We investigate with local constraints, namely, we assume that the set edges incident to vertex large degree not monochromatic. graph nice has no comp...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...