On The Spectral Charaterization Of The Line Graph Of T-shape Trees

In recent years，the research on spectral theory of graph is an active and very importantresearch fields. One of the main researcch direction is the algebraic properties of graphs,that is the matrix eigenvalue，such as, spectral radius, the only certainty spectrum and soon. The research on spectral theory of graph has wide applications in many fields, suchas quantum chemistry, statistical mechanics, computer science, network optimization anddesign, integrated circuit design and operations research and so on.In order to research the algebraic properties of graphs and spectrum of a matrix, thepeople usually apply three kinds of spectra in the theory of graph, such as, the spectra ofadjacent matrix(A-spectra), the spectra of Laplacian matrix(L-spectra) and the spectra ofsinless Laplacian matrix(Q-spectra). Up to now, a number of previously results about thespectral characterization of the unicycle graph have established. In[39], W illemH.Haemersproved that T (a, b, c) is determined by its adjacency spectrum and Laplacian spectrum; in[26],Y uanping Zhang showed that T (a, b, c) is determined by its signless Laplacian spectrum;in[33], Jian feng W ang has discussed the adjacency spectral characterization of the linegraph of Lollipop; in[40]，Guangquan Guo and Guoping Wang has discussed the (sign-less) Laplacian spectral characterization of the line graph of Lollipop; in[24][25], W ei W angshowed that T-shape trees T (a, b, c) is DLS and DAS; in[21], G.R. Omidi has discussedthe Q-spectral characterization of the Lollipop and the A-spectral characterization of theline graph of Lollipop.The paper consists of two sections. In the first section, it showed that all L(T (1, b, c))are DLS; In the second section, it showed that all L(T (a, b, c)) but L(T (t, t,2t+1))(t≥1)are DQS, Furthermore, we give the limit of ρ1(L(T (a, b, c))).