# A spectral characterization of the $s$-clique extension of the triangular graphs

## TL;DR

This paper proves that for large enough n, co-edge-regular graphs cospectral with the s-clique extension of triangular graphs are uniquely characterized as such extensions, revealing a spectral characterization of these graph classes.

## Contribution

It establishes a spectral characterization for the s-clique extension of triangular graphs, showing uniqueness for large n among co-edge-regular graphs.

## Key findings

- Co-edge-regular graphs cospectral with s-clique extensions are uniquely identified as such for large n.
- Spectral properties determine the structure of s-clique extensions of triangular graphs.
- The result applies for all integers s ≥ 2 and sufficiently large n.

## Abstract

A regular graph is co-edge regular if there exists a constant $\mu$ such that any two distinct and non-adjacent vertices have exactly $\mu$ common neighbors. In this paper, we show that for integers $s\ge 2$ and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.