Some families of integral mixed graphs

Abstract

A mixed graph $\hat{G}$ is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix , $\hat{A}(\hat{G})$, of a mixed graph has rows and columns indexed by the set of vertices of $\hat{G}$, 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 \textit{mixed asymmetric product of mixed graphs} is introduced, and the notions of \textit{joining by arcs and joining by edges} are used, allowing us to obtain a new integral mixed graph from two original integral mixed graphs.