The diameter of the acyclic Birkhoff polytope

In this work we give an interpretation of vertices and edges of the acyclic Birkhoff polytope, , where T is a tree with n vertices, in terms of graph theory. We generalize a recent result relatively to the diameter of the graph .http://www.sciencedirect.com/science/article/B6V0R-4R70RHM-1/1/4f38cb080e47b5fa8e0d6c36588d41a

On generalized binomial series and strongly regular graphs

We consider a strongly regular graph, G, and associate a three dimensional Euclidean Jordan algebra, V, to its adjacency matrix A. Then, by considering binomial series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of V associated to A, we establish feasibility conditions for the existence of strongly regular graphs

On Randic Spread

A new spectral graph invariant sprR , called Randíc spread, is defined and investigated. This quantity is equal to the maximal difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius. Lower and upper bounds for sprR are deduced, some of which depending on the Randíc index of the underlying graph

Feasibility conditions on the parameters of a strongly regular graph

We consider a strongly regular graph, G, and associate a three dimensional Euclidean Jordan algebra, V, to the adjacency matrix A of G. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of V, we establish new feasibility conditions for the existence of strongly regular graphs

Faces of faces of the tridiagonal Birkhoff polytope

AbstractThe tridiagonal Birkhoff polytope, Ωnt, is the set of real square matrices with nonnegative entries and all rows and columns sums equal to 1 that are tridiagonal. This polytope arises in many problems of enumerative combinatorics, statistics, combinatorial optimization, etc. In this paper, for a given a p-face of Ωnt, we determine the number of faces of lower dimension that are contained in it and we discuss its nature. In fact, a 2-face of Ωnt is a triangle or a quadrilateral and the cells can only be tetrahedrons, pentahedrons or hexahedrons

Face counting on an Acyclic Birkhoff polytope

http://www.sciencedirect.com/science/article/B6V0R-4V5GD2H-1/2/5a4ab2f05e5a55a500bf0fc93554003

On the number of invariant polynomials of matrix commutators

We study the possible numbers of nonconstant invariant polynomials of the matrix commutator XA-AX; when X varies

Eigenvalues of Matrix commutators

We characterize the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size

On the spectra of some graphs like weighted rooted trees

AbstractLet G be a weighted rooted graph of k levels such that, for j∈{2,…,k}(1)each vertex at level j is adjacent to one vertex at level j-1 and all edges joining a vertex at level j with a vertex at level j-1 have the same weight, where the weight is a positive real number;(2)if two vertices at level j are adjacent then they are adjacent to the same vertex at level j-1 and all edges joining two vertices at level j have the same weight;(3)two vertices at level j have the same degree;(4)there is not a vertex at level j adjacent to others two vertices at the same level;We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix of G. They are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k×k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices. By application of the above mentioned results, we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached, by one of its vertices, to a pendant vertex of the tree

On complementary coverage of Ωn(T)

Every acyclic Birkhoff polytope is represented by a bicolored tree. In this paper we use the concept of T-component of a tree in order to cover it. In addition, the definitions of T-edge cover(respectively, T-vertex cover)subgraphs and of complementary coverage by vertices (edges) are introduced. Some consequences related to the dimension of the acyclic Birkhoff polytope are also obtained