On bipartite graphs of diameter 3 and defect 2

Abstract

We consider bipartite graphs of degree DZ2, diameter D 5 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (D, 3, À2) -graphs. We prove the uniqueness of the known bipartite (3, 3, À2) -graph and bipartite (4, 3, À2)-graph. We also prove several necessary conditions for the existence of bipartite (D, 3, À2)graphs. The most general of these conditions is that either D or DÀ2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when D 5 6 and D 5 9, we prove the non-existence of the corresponding bipartite (D, 3, À2)-graphs, thus establishing that there are no bipartite (D, 3, À2)-graphs, for 5rDr10.Moreover, for D odd, by using a more careful counting argument, we obtain Proposition 1.2.Proposition 1.2. For do2½ðD À 1Þ þ ðD À 1Þ 3 þ Á Á Á þ ðD À 1Þ DÀ2 , DZ3 and odd DZ3, a bipartite (D, D, Àd)-graph is regular.

Non-existence of bipartite graphs of diameter at least 4 and defect 2

“…Henceforth, let λ 1 < λ 2 < · · · < λ r be the roots of H r (x) − 1 = 0, and let ρ 1 < ρ 2 < · · · < ρ r be the roots of H r (x) + 1 = 0. Consequently, it suffices to prove that, for any pair (Δ, r) other than (3,5), (3,6), (3,7), (3,8), This completes the proof of the lemma.…”

“…The fact that rep is an automorphism of Γ was proved in [8]. The permutation matrix associated with rep is called the defect matrix of Γ and plays an important role in the study of the structure of Γ (see [7]).…”

“…Therefore, considering (7) and (8), it suffices to show that, for Δ = 3, 4 and r ≥ 5, with the exceptions of the pairs (Δ, r) = (3, 5), (3,6), (3,7), and (3,8), the following inequality holds:…”

“…With the exception of N b Conditions for under which a bipartite (Δ, D, − )-graph must be regular of degree Δ were established in [8], and one of them is stated below. In this paper we analyze the case of defect 2.…”

“…Concerning bipartite (Δ, D, −2)-graphs, the paper [8] considered bipartite (Δ, 3, −2)-graphs, deriving some necessary conditions for their existence, and proving the uniqueness of the two known bipartite (Δ, 3, −2)-graphs (both graphs are depicted in Fig. 1).…”