Some algebraic properties of a class of integral graphs determined by their spectrum

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph $\Gamma=Cay(\mathbb{Z}_{n}, S)$, where $n=p^m$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\, (a, n)=1\}$. First, we show that $\Gamma$ is an integral graph. Also we determine the automorphism group of $\Gamma$. Moreover, we show that $\Gamma$ and $K_v \bigtriangledown\Gamma$ are determined by their spectrum

Could the 21-cm absorption be explained by the dark matter suggested by 8^8Be transitions?

The stronger than expected 21-cm absorption was observed by EDGES recently, and another anomaly of $^8$Be transitions would be signatures of new interactions. These two issues may be related to each other, e.g., pseudoscalar $A$ mediated fermionic millicharged dark matter (DM), and the 21-cm absorption could be induced by photon mediated scattering between MeV millicharged DM and hydrogen. This will be explored in this paper. For fermionic millicharged DM $\bar{\chi} \chi$ with masses in a range of $2 m_A < 2 m_{\chi} < 3 m_A$, the p-wave annihilation $\bar{\chi} \chi \to A A$ would be dominant during DM freeze-out. The s-wave annihilation $\bar{\chi} \chi$ $\to A, \gamma$ $\to e^+ e^-$ is tolerant by constraints from CMB and the 21-cm absorption. The millicharged DM can evade constraints from direct detection experiments. The process of $K^+ \to \pi^+ \pi^0$ with the invisible decay $\pi^0 \to \bar{\chi} \chi$ could be employed to search for the millicharged DM, and future high intensity $K^+$ sources, such as NA62, will do the job.Comment: 6 pages, 2 figures, the accepted version, EPJ

Some resolving parameters in a class of Cayley graphs

Resolving parameters is a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this article, we construct a class of Toeplitz graphs, and will be denoted by $T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs are vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and since the computing resolving parameters of a class of graphs such that are not distance regular is more difficult, then we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set and strong metric dimension for this class of graphs.Comment: arXiv admin note: substantial text overlap with arXiv:1905.1052