On Graphs with Cyclic Defect or Excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ǫ are called graphs with defect or excess ǫ, respectively. While Moore graphs (graphs with ǫ = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation Gd,k(A) = Jn+B (Gd,k(A) = Jn−B), where A denotes the adjacency matrix of the graph in question, n its order, Jn the n× n matrix whose entries are all 1’s, B the adjacency matrix of a union of vertex-disjoint cycles, and Gd,k(x) a polynomial with integer coefficients such that the matrix Gd,k(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O( 3 d 3/2) for the number of graphs of odd degree d > 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d > 3 and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree > 3 and cyclic defect or excess exists.