Let G be a nontrivial graph. A set D ⊆ V(G) is a double dominating set of G if | \mathrm{N_G} [v] ∩ D| ≥ 2 for every vertex v ∈ V(G), where \mathrm{N_G} [v] represents the closed neighborhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this paper we...
This work establishes general properties of k -total bondage and finds exact values for certain graph classes including paths, cycles, wheels, complete and complete bipartite graphs.
An independent set of a graph G is a subset of VG, no two of which are adjacent. The cardinality of a maximum independent set in a graph G is called the independence number of G, denoted by alpha(G). The essential connectivity kappa'(G) of a graph G is denoted as the minimum number of vertices of G whose removal produc...
A connected graph, whose blocks are all cliques (of possibly varying sizes), is called a block graph. Let $D(G)$ be its distance matrix. In this note, we prove that the Smith normal form of $D(G)$ is independent of the interconnection way of blocks and give an explicit expression for the Smith normal form in the case t...
Let $s^+(G)$ denote the sum of the squares of the positive adjacency eigenvalues of a graph $G$. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan, proved by Liu, Tang, and Zhang, gives a lower bound of $n-1$ for any connected graph of order $n$. We strengthen this bound to $s^+(G)\ge n$ for every c...
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