On Bipartite Cages of Excess 4

The Moore bound M(k, g) is a lower bound on the order of k-regular graphs of girth g (denoted (k, g)-graphs). The excess e of a (k, g)-graph of order n is the difference n −M(k, g). In this paper we consider the existence of (k, g)-bipartite graphs of excess 4 by studying spectral properties of their adjacency matrices. For a given graph G and for the integers i with 0 6 i 6 diam(G), the i-distance matrix Ai of G is an n× n matrix such that the entry in position (u, v) is 1 if the distance between the vertices u and v is i, and zero otherwise. We prove that the (k, g)bipartite graphs of excess 4 satisfy the equation kJ = (A + kI)(Hd−1(A) + E), where A = A1 denotes the adjacency matrix of the graph in question, J the n× n all-ones matrix, E = Ad+1 the adjacency matrix of a union of vertex-disjoint cycles, and Hd−1(x) is the Dickson polynomial of the second kind with parameter k−1 and degree d − 1. We observe that the eigenvalues other than ±k of these graphs are roots of the polynomials Hd−1(x) + λ, where λ is an eigenvalue of E. Based on the irreducibility of Hd−1(x)± 2, we give necessary conditions for the existence of these graphs. If E is the adjacency matrix of a cycle of order n, we call the corresponding graphs graphs with cyclic excess; if E is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of (k, g)-graphs with cyclic excess 4 if k > 6 and k ≡ 1(mod 3), g = 8, 12, 16 or k ≡ 2(mod 3), g = 8; and the non-existence of (k, g)-graphs with bicyclic excess 4 if k > 7 is an odd number and g = 2d such that d > 4 is even.