Some families of integral mixed graphs

Abstract

A mixed graph Gˆ is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix, Aˆ(Gˆ), of a mixed graph has rows and columns indexed by the set of vertices of Gˆ, being its {u,v}-entry equal to 1 (respectively, −1) if the vertex u is connected by an edge (respectively, an arc) to the vertex v, and 0 otherwise. These graphs are called integral mixed graphs if the eigenvalues of its adjacency matrix are integers. In this paper, symmetric block circulant matrices are characterized, and as a consequence, the definition of a mixed graph to be a block circulant graph is presented. Moreover, using this concept and the concept of a g-circulant matrix, the construction of a family of undirected graphs that are integral block circulant graphs is shown. These results are extended using the notion of H-join operation to characterize the spectrum of a family of integral mixed graphs. Furthermore, a new binary operation called mixed asymmetric product of mixed graphs is introduced, and the notions of joining by arcs and joining by edges are used, allowing us to obtain a new integral mixed graph from two original integral mixed graphs.