AN UPPER BOUND ON ALGEBRAIC CONNECTIVITY OF GRAPHS WITH MANY CUTPOINTS

Let G be a graph on n vertices which has k cutpoints. A tight upper bound on the algebraic connectivity of G in terms of n and k for the case that k>n /2 is provided; the graphs which yield equality in the bound are also characterized. This completes an investigation initiated by the author in a previous paper, which dealt with the corresponding problem for the case that k ≤ n/2. G). Finally, if G has vertex connectivity c ≤ n − 2, then α(G) ≤ c .I n particular, if G has a cutpoint - that is, a vertex whose deletion (along with all edges incident with it) yields a disconnected graph - then we see that α(G) ≤ 1. Motivated by this last observation, Kirkland (7) posed the following problem: if G is a graph on n vertices which has k cutpoints, find an attainable upper bound on α(G). In (7), such a bound is constructed for the case that 1 ≤ k ≤ n/2, and the graphs attaining the bound are characterized. The present paper is a continuation of the work in (7); here we give an attainable upper bound on α(G )w henn/2 <k ≤ n−2, and explicitly describe the equality case. The technique used in this paper relies on the analysis of the various connected components which arise from the deletion of a cutpoint. We now briefly outline that technique. Suppose that G is a connected graph and that v is a cutpoint of G.Th e components at v are just the connected components of G−v, the (disconnected) graph ∗ Received by the editors on 2 November 2000. Accepted for publication on 19 May 2001. Handling