Bipartite graphs with five eigenvalues and pseudo designs

TL;DR

This paper characterizes certain bipartite graphs with five eigenvalues using pseudo designs and provides partial evidence supporting a conjecture on their structure.

Contribution

It introduces a spectral characterization of graphs via pseudo designs and advances the understanding of their eigenvalue properties.

Findings

Graphs with specific eigenvalue multiplicities are characterized by pseudo designs.

Partial results support Marrero's conjecture on pseudo design structures.

Spectral properties relate to combinatorial design configurations.

Abstract

A pseudo $(v,\, k,\, \la)$-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in $\la$ elements; and $0\le\la <k \le v-1$. We use the notion of pseudo designs to characterize graphs of order $n$ whose (adjacency) spectrum contains a zero and $\pm\theta$ with multiplicity $(n-3)/2$ where $0<\theta\le\sqrt{2}$. Meanwhile, partial results confirming a conjecture of O. Marrero on characterization of pseudo $(v,\, k,\, \la)$-designs are obtained.