Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies...

Abstract Let A and B be n×n complex matrices. Characterization is given for the set E(A,B) of eigenvalues of matrices of the form UAU + V BV for some unitary matrices U and V . Consequences of the results are discussed and computer algorithms and programs are designed to generate the set E(A,B). The results refine those of Wielandt on normal matrices. Extensions of the results to the sum of mat...

We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “(3,6)-fullerenes”, have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form {λ,−λ} except for the four eigenvalues {3,−1,−1,−1}. We exhibit other families of graphs which ar...

A graph Γ with diameter d is strongly distance-regular if Γ is distanceregular and its distance-d graph Γd is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with...

The P1 discretization of the Laplace operator on a triangulated polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the P1 discretization of the Laplace operator. Among all triangles, an equilateral triangle has the maximal first positive eigenvalue. Among all cyclic quadrilateral, a square has the maximal first positive ...