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

Abstract

We confirm the following conjecture which has been proposed in [{\em Linear Algebra and its Applications}, {\bf 436} (2012), No. 5, 1425-1435.]: $$0.945\approx \underset{n⟶\infty }{\lim }\sigma (P_n,Z_n)=\underset{n⟶\infty }{\lim }\sigma (W_n,Z_n)=\frac{1}{2}\underset{n⟶\infty }{\lim }\sigma (P_n,W_n);\underset{n⟶\infty }{\lim }\sigma (C_{2n},Z_{2n})=2,$$ 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 ${\lambda }_1(G_i)\ge {\lambda }_2(G_i)\ge \cdots \ge {\lambda }_n(G_i)$ 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 $P_3$.