H-integral normal mixed Cayley graphs

A mixed graph is called integral if all the eigenvalues of its Hermitian adjacency matrix are integers. A mixed Cayley graph $Cay(\Gamma, S)$ is called normal if $S$ is the union of some conjugacy classes of a finite group $\Gamma$. In 2014, Godsil and Spiga characterized integral normal Cayley graphs. We give similar characterization for the integrality of a normal mixed Cayley graph $Cay(\Gamma,S)$ in terms of $S$. Xu and Meng (2011) and Li (2013) characterized the set $S\subseteq \mathbb{Z}_n$ for which the eigenvalues $\sum\limits_{k\in S} w_n^{jk}$ of the circulant digraph $Cay(\mathbb{Z}_n, S)$ are Gaussian integers for all $j=1,...,h$. Here the adjacency matrix of $Cay(\mathbb{Z}_n, S)$ is considered to be the $n\times n$ matrix $[a_{ij}]$, where $a_{ij}=1$ if $(i,j)$ is an arc of $Cay(\mathbb{Z}_n, S)$, and $0$ otherwise. Let $\{\chi_1,\ldots,\chi_h\}$ be the set of the irreducible characters of $\Gamma$. We prove that $\frac{1}{\chi_j(1)} \sum\limits_{s \in S} \chi_j(s)$ is a Gaussian integer for all $j=1,...,h$ if and only if the normal mixed Cayley graph $Cay(\Gamma, S)$ is integral. As a corollary to this, we get an alternative and easy proof of the characterization, as obtained by Xu, Meng and Li, of the set $S\subseteq \mathbb{Z}_n$ for which the circulant digraph $Cay(\mathbb{Z}_n, S)$ is Gaussian integral

Small integral trees

We give a table with all integral trees on at most 50 vertices, and characterize integral trees with a single eigenvalue 0. 1 Integral trees A finite graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues. A tree is a connected undirected graph without cycles. In this note we give a table with all integral trees on at most 50 vertices, and a further table with all known integral trees on at most 100 vertices. (For an on-line version, possibly with updates, see [1].) In particular, we find the smallest integral trees of diameter 6, and the smallest known integral tree of diameter 8. The nicest result about integral trees is that by Watanabe [12] that says that an integral tree different from K2 does not have a complete matching. Here we give a generalization