Unordered Multiplicity Lists Of A Class Of Trees

The spectrum of a graph G is the set of the spectra δ (A)of all matrices A isS (G).The (n-1)×(n-1) principle submatrix, formed by the deletion of rowand column indexed by i, which is equivalent to removing the vertex i fromG, is designated by A (G\i).Let us denote the (algebraic) multiplicity of theeigenvalue θ of a symmetric matrix A=A(G)by m A(θ). Ifm1≥m2≥≥mkare the multiplicities of the distinct eigenvalues of matrix A,then the list(m1,m2,,mk)is called the unordered multiplicity list of the eigenvalues of A.For example, if a6×6matrix A has eigenvalues2,4,4,4,7,7, then the unorderedmultiplicity list of the eigenvalues of A is (3,2,1).Letli denote the length of the pathPi.v1andv2are the end vertices of thepathv1v v2,then we add pathsP1,P2tov1,and add pathsP3,P4tov2,wherel1≥l3≥l4,l1≥l2.We call the tree like this as Φ-binary tree.We carry this discussion forward extending their results to a larger family oftrees. It consists of seven pathsP1,P2,P3,P4,P5,P6,P7,and two vertices u, v. Thetree is formed as follows: join any terminal vertex ofp1,p2,p3,p7to u;and theother terminal vertex ofp7and the terminal vertex ofp4,p5,p6are added to v.The pathsp1,p2,p3,p4,p5,p6are the legs of such trees.