A conjecture about spectral distances between cycles, paths and certain trees

TL;DR

This paper proves a conjecture about the limiting spectral distances between specific graph families, including paths, cycles, and certain trees, confirming their asymptotic behavior as the number of vertices grows large.

Contribution

It confirms a conjecture on the asymptotic spectral distances between paths, cycles, and related trees, providing precise limit values and relations.

Findings

Spectral distance between paths and certain trees approaches 0.945.

Spectral distance between cycles and a specific tree approaches 2.

Spectral distances satisfy particular limits as graph size tends to infinity.

Abstract

We confirm the following conjecture which has been proposed in [{\em Linear Algebra and its Applications}, {\bf 436} (2012), No. 5, 1425-1435.]: $$ 0.945\approx\displaystyle\lim_{n\longrightarrow \infty}\sigma(P_n,Z_n)=\displaystyle\lim_{n\longrightarrow \infty}\sigma(W_n,Z_n)=\frac{1}{2}\displaystyle\lim_{n\longrightarrow \infty}\sigma(P_n,W_n);\ \displaystyle\lim_{n\longrightarrow \infty}\sigma(C_{2n},Z_{2n})=2,$$ where $\sigma(G_1,G_2)=\sum_{i=1}^n |\lambda_i(G_1)-\lambda_i(G_2)|$ is the spectral distance between $n$ vertex non-isomorphic graphs $G_1$ and $G_2$ with adjacency spectra $\lambda_1(G_i) \geq \lambda_2(G_i) \geq \cdots \geq \lambda_n(G_i)$ for $i=1,2$, and $P_n$ and $C_n$ denote the path and cycle on $n$ vertices, respectively; $Z_n$ denotes the coalescence of $P_{n-2}$ and $P_3$ on one of the vertices of degree 1 of $P_{n-2}$ and the vertex of degree $2$ of $P_3$; and $W_n$ denotes the coalescence of $Z_{n-2}$ and $P_3$ on the vertex of degree 1 of $Z_{n-2}$ which is adjacent to a vertex of degree $2$ and the vertex of degree $2$ of $P_3$.