Zero forcing is an iterative graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. It is NP-hard to find a minimum zero forcing set – a smallest set of initially colored vertices which forces the entire graph to be colored. We show that the problem remains NP-hard when the initially colored set induces a connected subgraph. We also ...

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n 2 ). In this paper we discuss the ...

Let G be a simple graph with vertex set V (G). A non-empty set S ⊆ V (G) is a global strong offensive alliance if for every vertex v in V (G)−S, a strict majority of its closed neighborhood is in S. The global strong offensive alliance number γô(G) is the minimum cardinality of a global strong offensive alliance of G. We show that if G is a connected bipartite graph of order at least three, the...

In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that any pair graph, there is vertex from whose two distances to are distinct. This problem has mainly been studied on undirected graphs and gained lot attention in recent years. We focus directed graphs, show how solve linear-time digraphs underlying graph (ignoring multiple edges) tree. (nontrivially) extends ...

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.