Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...

Let −→ G be a digraph and A( −→ G) be the adjacency matrix of −→ G . Let D( −→ G) be the diagonal matrix with outdegrees of vertices of −→ G and Q( −→ G) = D( −→ G) + A( −→ G) be the signless Laplacian matrix of −→ G . The spectral radius of Q( −→ G) is called the signless Laplacian spectral radius of −→ G . In this paper, we determine the unique digraph which attains the maximum (or minimum) s...

Abstract We derive an analytical expression for the mean load at each node of arbitrary undirected graph non-uniform multicommodity flow problem under weighted random routing. show node, net its demand and normalized by (weighted) degree, is a constant equal to trace product two matrices: Laplacian matrix generalized inverse Laplacian. For case uniform demand, this reduces sum inverses non-zero...

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the 21 connected Laplacian integral graphs of maximum degree 3 on at least 6 vertices.

Let G be a graph; its Laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. In this paper, we present a sharp upper bound for the Laplacian spectral radius of a tree in terms of the matching number and number of vertices, and deduce from that the largest few Laplacian spectral radii over the class of trees on a given number of vertices. © 2003...

In this paper, all connected bipartite graphs are characterized whose third largest Laplacian eigenvalue is less than three. Moreover, the result is used to characterize all connected bipartite graphs with exactly two Laplacian eigenvalues not less than three, and all connected line graphs of bipartite graphs with the third eigenvalue of their adjacency matrices less than one. c © 2003 Elsevier...

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

Abstract The eccentricity matrix ? ( G ) of a graph is obtained from the distance by retaining largest distances in each row and column, leaving zeros remaining ones. energy sum absolute values eigenvalues ). Although matrices graphs are closely related to graphs, number properties substantially different those matrices. change due an edge deletion one such property. In this article, we give ex...