Spectral characterization of the complete graph removing a path of small length

TL;DR

This paper investigates whether certain graphs obtained by removing a small path from a complete graph are uniquely identified by their spectra, confirming the conjecture for path lengths 7 to 9.

Contribution

The paper proves the spectral determination conjecture for graphs formed by removing a path of length 7 to 9 from a complete graph.

Findings

Confirmed the conjecture for 7 ≤ ℓ ≤ 9

Spectral characterization holds for these specific path lengths

Supports broader conjecture on spectral uniqueness

Abstract

A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a path of length $\ell-1$ which is a subgraph of a complete graph $K_n$. C\'{a}mara and Haemers~\cite{MC} conjectured that $K_n \backslash P_{\ell}$ is determined by its adjacency spectrum for every $2\leq \ell \leq n$. In this paper we show that the conjecture is true for $7\leq \ell \leq9$.