# On the spectrum of two families of non-distance-regular graphs

## TL;DR

This paper develops a novel spectral analysis method for non-distance-regular graphs, successfully determining all eigenvalues and their multiplicities for specific classes, including the extended graph $E(2.O_k)$ and the enhanced Johnson graph $EJ(2m,m)$.

## Contribution

It introduces a new approach leveraging equitable and orbit partitions to find complete eigenvalue spectra of non-distance-regular graphs, including multiplicities.

## Key findings

- Eigenvalues of $E(2.O_k)$ are all integers.
- Determined the automorphism group of $E(2.O_k)$.
- Calculated the multiplicities of eigenvalues for $EJ(2m,m)$.

## Abstract

This paper addresses the challenge of spectral analysis and structural investigation for graphs that are not distance-regular, where computing the spectrum using standard methods based on equitable and orbit partitions can be complex. Our main objective is to determine all eigenvalues of the extended graph $E(2.O_k)$ by leveraging the relationship between its equitable and orbit partitions. While the integral nature of this graph has been previously studied, we introduce a novel approach to demonstrate the utility of this method in finding the complete set of distinct eigenvalues for a class of non-distance-regular graphs. Specifically, we first establish that $E(2.O_k)$ is a vertex-transitive graph with diameter $k$, contrasting with the diameter of $2.O_k$, which is $2k-1$. We also determine the automorphism group of $E(2.O_k)$ and prove that it is an integral graph, meaning all eigenvalues of its adjacency matrix are integers. A significant result is the determination of the multiplicity for all distinct eigenvalues of $E(2.O_k)$. Additionally, we extend our method to the enhanced Johnson graph $EJ(2m,m)$. Although its eigenvalues are known from prior work, the multiplicity of these distinct eigenvalues has not yet been calculated. We use our techniques to fully determine the multiplicity of all distinct eigenvalues for $EJ(2m,m)$.