Conjectures on index and algebraic connectivity of graphs

Abstract

Let G = ( V , E ) be a simple graph with vertex set V = { v 1 , v 2 , … , v n } and edge set E ( G ) . The adjacency matrix of a graph G is denoted by A ( G ) and defined as the n × n matrix ( a ij ) , where a ij = 1 for v i v j ∈ E ( G ) and 0 otherwise. The largest eigenvalue ( λ 1 ) of A ( G ) is called the spectral radius or the index of G . The Laplacian matrix of G is L ( G ) = D ( G ) - A ( G ) , where D ( G ) is the diagonal matrix of its vertex degrees and A ( G ) is the adjacency matrix. Among all eigenvalues of the Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic connectivity ( a ) of a graph [12]. In [1,2], Aouchiche et al. have given a series of conjectures on index ( λ 1 ) and algebraic connectivity ( a ) of G (see also [3]). Here we prove two conjectures and disprove one by a counter example.