On the Laplacian Spectral Characterization of the Generalized $T$-Shape Trees

Abstract

A $T$-shape tree $T(l_1,l_2,l_3)$ is obtained from three paths $P_{l_1+1}$, $P_{l_2+1}$, and $P_{l_3+1}$ by identifying one of their pendent vertices. A generalized $T$-shape tree $T_s(l_1,l_2,l_3)$ is obtained from $T(l_1,l_2,l_3)$ by appending two pendent vertices to exactly $s$ pendent vertices of $T(l_1,l_2,l_3)$, where $1\le s\le 3$ is a positive integer. In this paper, we firstly show that the generalized $T$-shape tree $T_2(l_1,l_2,l_3)$ is determined by its Laplacian spectrum. Applying similar arguments for the trees $T_1(2l_1,l_2,l_3)$ and $T_3(l_1,2l_2,l_3)$, one can obtain that any generalized $T$-shape tree on $n$ vertices is determined by its Laplacian spectrum.