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 \(1 \leq s \leq 3\) 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.