On graphs with equal algebraic and vertex connectivity

Let G 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 L(L-#) := (1/2)max(1less than or equal toi, jless than or equal ton) Sigma(s)(n)=1 \l(i,s)(#) - l(j,s)(#)\, the quantity 1/ L(L-#) is a lower bound on the algebraic connectivity a (G) of G, while the vertex connectivity of G, v(G), is an upper bound on a (G). We characterize the graphs G for which v(G) = a(G) and subsequently prove that if n greater than or equal to v(G)(2), then v(G) = a(G) holds if and only if 1/L(L-#) = a(G) = v(G). We close with an example showing that the equality 1/L(L-#) = a(G) does not necessarily imply that 1/L(L-#) = a(G) = v(G). (C) 2002 Elsevier Science Inc. All rights reserved.