Let γ(G) and #γ(G) denote the domination number and the number of all distinct minimum dominating sets of a graph G, respectively. We show that #γ(G+ e) ≥ #γ(G) for every edge e ∈ E(G) with γ(G+e) = γ(G). Based on this, we begin an investigation of graphs for which #γ increases whenever an edge is added (we call such graphs #γ-EA-critical). We prove that if G is #γ-EA-critical then either G is ...

In this paper we will apply the tensor and its traces to investigate spectral characterization of unicyclic graphs. Let $G$ be a graph $G^m$ $m$-th power (hypergraph) $G$. The spectrum is referring adjacency matrix, tensor. called determined by high-ordered spectra (DHS for short) if, whenever $H$ such that $H^m$ cospectral with all $m$, then isomorphic first give formulas graphs, provide some ...

By d(v|G) and d_2(v|G) are denoted the number of first and second neighborsof the vertex v of the graph G. The first, second, and third leap Zagreb indicesof G are defined asLM_1(G) = sum_{v in V(G)} d_2(v|G)^2, LM_2(G) = sum_{uv in E(G)} d_2(u|G) d_2(v|G),and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(v|G), respectively. In this paper, we generalizethe results of Naji et al. [Commun. Combin. Optim. ...

Connected graphs in which the number of edges equals the number of vertices are called unicyclic graphs. In this paper, all unicyclic graphs whose second largest eigenvalue does not exceed 1 have been determined. ? 2003 Elsevier B.V. All rights reserved.

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 ...

Recently, the first and second Zagreb indices are generalized into the variable Zagreb indices which are defined by M1(G) = ∑ u∈V (d(u))2λ and M2(G) = ∑ uv∈E (d(u)d(v)), where λ is any real number. In this paper, we prove that M1(G)/n M2(G)/m for all unicyclic graphs and all λ ∈ (−∞, 0]. And we also show that the relationship of numerical value between M1(G)/n and M2(G)/m is indefinite in the d...

In an ordinary edge-coloring of a graph each color appears at each vertex at most once. An f -coloring is a generalized edge-coloring in which each color appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f -colorings for various classes of graphs such as bi...

Topological indices play an important role in Mathematical Chemistry especially in the quantitative structure-property relationship (QSPR) and quantitative structureactivity relationship (QSAR). Recent research indicates that the augmented Zagreb index (AZI) possess the best correlating ability among several topological indices. The main purpose of the current study is to establish some mathema...

The Wiener polarity indexWP (G) of a graph G is the number of unordered pairs of vertices {u, v} of G such that the distance of u and v is equal to 3. In this paper, we obtain the relation between Wiener polarity index and Zegreb indices, and the relation between Wiener polarity index and Wiener index (resp. hyper-Wiener index). Moreover, we determine the second smallest Wiener polarity index t...