On graphs with equal algebraic and vertex connectivity Original Research Article

Let image be an undirected unweighted graph on n vertices, let L be its Laplacian matrix, and let L#=(ℓ#i,j) be the group inverse of L. It is known that for image the quantity image is a lower bound on the algebraic connectivity image of image , while the vertex connectivity of image , image , is an upper bound on image . We characterize the graphs image for which image and subsequently prove that if image , then image holds if and only if image . We close with an example showing that the equality image does not necessarily imply that image