Two Distinct Eigenvalues from a New Graph Product

TL;DR

This work studies the graph parameter $q(G)$, the minimum number of distinct eigenvalues across symmetric matrices with a fixed graph pattern, focusing on graphs where $q(G)=2$. It introduces a modified strong product $G \Bowtie H$ with adjacency $A_G \otimes (A_H+I)$ and develops a robust tensor-product framework: a carefully chosen orthogonal $Q$ yields idempotent blocks $J_i$ that decompose spectra, and a main lemma shows that $M=\sum_i (A_i \otimes J_i)$ inherits eigenpairs from the factors. Using edge-colorings and hypergraph representations, the paper constructs broad infinite families with $q(G)=2$, including $G \Bowtie K_k$ when $G$ is $k$-regular with $\chi'(G)=k$, $G \otimes K_{k+1}$ for $\Delta(G)=k$, and hypergraph-based products yielding similar results for other regimes. This unifies several known $q(G)=2$ examples under a common product-based framework and opens avenues for generalizations and characterizations of graphs arising from these constructions.

Abstract

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues of a symmetric matrix whose pattern is given by $G$. We introduce a novel graph product by which we construct new infinite families of graphs that achieve $q(G)=2$. Several graph families for which it is already known that $q(G)=2$ can also be thought of as arising from this new product.

Figures (3)

• Figure 1: Two examples of modified strong products. Note in particular that $K_2\ \raisebox{1pt}{origin=c]{90}{\Bowtie}}\; P_3$ is not isomorphic to $P_3\ \raisebox{1pt}{origin=c]{90}{\Bowtie}}\; K_2$.
• Figure 2: Some examples of graphs where Theorem \ref{['thm:complete']} applies.
• Figure 3: Modified strong products of cycles and paths with $K_2$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof