The sharpness of a lower bound on the algebraic connectivity for maximal graphs

Let B be an undirected unweighted graph on n vertices and let L be its Laplacian matrix. It is known that if L-# = (e(ij)((#))) is the group inverse of L, then for Z(L-#) := (1/2) max(1 less than or equal toi,j less than or equal ton) Sigma (n)(s=1) /l(i,s)((#)) - l(j,s)((#))/, is a lower bound on the algebraic connectivity mu (G) Of S. Merris has introduced and characterized the class of all maximal graphs of all orders. These are graphs whose degree sequence is not majorized by the degree sequence of any other graph. Here we show that if G is a maximal graph and L is its Laplacian, then 1/Z(L-#) = mu (G). We provide an example to show that the converse of this result is not valid.