A spectral version of the Moore problem for bipartite regular graphs

Abstract

Let $b(k,\theta )$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta )$ for any $0\le \theta <2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta )$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, diameter $d$ and girth $g$ such that $g\ge 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\ge 15$, we prove that there are no bipartite distance-regular graphs with $g\ge 2d-2$.

24 pages, 7 tables; In the revised version, a new result (Theorem 4.12) is added at the end of Section 4 showing that the bipartite bound from this paper (Theorem 4.7) is always better than the general upper bound from our previous SIDMA paper (Theorem 4.9)