The Spectra Of Graphs Based On Join Operations And The Generalized Distance Spectrum

The spectral theory of graph is one of the most active project in the graph theory.It has important applications on theoretical chemistry,circuit theory and many aspect-s.Many variants of join operations of graphs have been introduced and their spectral properties have been studied extensively.At the same time.The spectral properties of distance matrix D(G)and distance signless Laplacian matrix DQ(G)of graph G have attracted much more attention.This paper do the study of the spectrum of graphs based on join operations and the generalized distance matrix mainly,and following results are obtained.For the Laplacian spectra of some double join operations of graphs.First,the concep-tion of double join matrix are introduced and complete information about its eigenvalues and the corresponding eigenvectors are provided.Further,four variants of double join operations based on subdivision graph,Q-graph,R-graph and total graph are defined.Applying the result obtained for the double join matrix,we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four vaxiants in terms of the Laplacian eigenvalues and the eigenvectors of the factor graphs.These results generalize some well-known results about L-spectrum of graphs based on join operations.The definition of S-edge-vertex graph based on join operation are given and their adjacency spectrum,Laplacian spectrum and normalized Laplacian spectrum in terms of the corresponding spectrum of the constituting graphs are determined.These results help to construct some classes of non-regular graphs.This work partly answers a problem proposed by Butler.In addition,the number of spanning trees,the Kirchhoff index and the degree-Kirchhoff index of constructed partial join are also obtained.The new convex combinations Dα(G)of Diag(Tr)and D(G)are studied.Dα(G)=αDiag(Tr)+(1-α)D(G),0≤α≤1.This paper call it the generalized distance matrix.Some spectral properties of Dα(G)are given and a few open problems are discussed.Furthermore,some upper and lower bounds of spectral radius of Da(G)are obtained.Finally,the generalized distance spectra of some graphs obtained by operations are also studied.