The topological indices, defined as the sum of contributions of all pairs of vertices (among which are the Wiener, Harary, hyper–Wiener indices, degree distance, and many others), are expressed in terms of contributions of edges and pairs of edges.

Let \(G=(V(G),E(G))\) be a simple graph. A set \(S \subseteq V(G)\) is strong edge geodetic if there exists an assignment of exactly one shortest path between each pair vertices from S, such that these paths cover all the edges E(G). The cardinality smallest number \(\mathrm{sg_e}(G)\) G. In this paper, problem studied on Cartesian product two paths. exact value computed for \(P_n \,\square \,P...

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor vertices and/or edges (viewed as graph) order detect and prevent failures. Inspired by two notions studied literature (edge-geodetic sets distance-edge-monitoring sets), we define notion monitoring edge-geodetic set (MEG-set for short) graph G an $$S\subseteq V(G)$$ (that is, every ...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...

ABSTRACT: A difference labeling of a graph G is realized by assigning distinct integer values to its vertices and then associating with each edge the absolute difference of those values assigned to its end vertices. A decomposition of labeled graph into parts, each part containing the edge having a commonweight is called a common – weight decomposition. In this paper we investigate the existenc...