On Bipartite Cages of Excess 4

  • Slobodan Filipovski
Keywords: Cage problem, Bipartite graphs, Cyclic excess, Bicyclic excess

Abstract

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 nM(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 0idiam(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)(Hd1(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 Hd1(x) is the Dickson polynomial of the second kind with parameter k1 and degree d1. We observe that the eigenvalues other than ±k of these graphs are roots of the polynomials Hd1(x)+λ, where λ is an eigenvalue of E. Based on the irreducibility of Hd1(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 k6 and k \equiv1 \!\! \pmod {3}, g=8, 12, 16 or k \equiv2 \!\! \pmod {3}, g=8; and the non-existence of (k,g)-graphs with bicyclic excess 4 if k\geq7 is an odd number and g=2d such that d\geq4 is even.

Published
2017-03-03
Article Number
P1.40