A CONJECTURE ON ALGEBRAIC CONNECTIVITY OF GRAPHS

Abstract

Let $G = (V, E)$ be a simple graph with vertex set $V(G) = \{v_1, v_2, \ldots, v_n\}$ and edge set $E(G)$. Let $A(G)$ be the adjacency matrix of graph $G$ and also let $D(G)$ be the diagonal matrix with degrees of the vertices on the main diagonal. The Laplacian matrix of $G$ is $L(G)=D(G)-A(G)$. Among all eigenvalues of the Laplacian matrix $L(G)$ of a graph $G$, the most studied is the second smallest, called the algebraic connectivity $(a(G))$ of a graph $G$ [9]. Let $\alpha(G)$ be the independence number of graph $G$. Recently, it was conjectured that (see, [1]): $$a(G)+\alpha(G)$$ is minimum for $\overline{K_{p,\,q}\backslash \{e\}}$, where $e$ is any edge in $K_{p,\,q}$ and $\displaystyle{p=\Big\lfloor\frac{n}{2}\Big\rfloor}\,,\, q=\Big\lceil\frac{n}{2}\Big\rceil$ ($K_{p,\,q}$ is a complete bipartite graph). The aim of this paper is to show that this conjecture is true.