A large family of cospectral Cayley graphs over dicyclic groups

For a finite group G and an inverse closed subset $S?G?\{e\}$, the Cayley graph $X(G,S)$ has vertex set G and two vertices $x,y\in G$ are adjacent if and only if $x{y}^{?1}\in S$. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let $p\ge 3$ be a prime number and $T_{4p}$ be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group $T_{4p}$ with $p\ge 23$, and find several pairs of non-isomorphic and cospectral Cayley graphs for $5\le p\le 19$.