Cartesian Prime Graphs and Cospectral Families

TL;DR

This paper presents a novel method for constructing large families of connected cospectral graphs using Cartesian primality, significantly expanding known cospectral families and their sizes.

Contribution

The authors introduce a new construction technique for cospectral graphs based on Cartesian primality, enabling the generation of exponentially larger families under various conditions.

Findings

Generated $O(p^3q^3)$ new cospectral triplets under strict conditions.

Produced $ ext{Omega}(pq^3 + qp^3)$ triplets with relaxed conditions.

Established the existence of larger cospectral families using their method.

Abstract

We introduce a method for constructing larger families of connected cospectral graphs from two given cospectral families of sizes $p$ and $q$. The resulting family size depends on the Cartesian primality of the input graphs and can be one of $pq$, $p + q - 1$, or $\max(p, q)$, based on the strictness of the applied conditions. Under the strictest condition, our method generates $O(p^3q^3)$ new cospectral triplets, while the more relaxed conditions yield $\varOmega(pq^3 + qp^3)$ such triplets. We also use the existence of specific cospectral families to establish that of larger ones.