arXiv subjects
Publications and source records attributed to Xueyi Huang.
Let $p$ be an odd prime, and $D_{2p}=\langle τ,σ\mid τ^p=σ^2=e,στσ=τ^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to isomorphism by using the Pólya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on $D_{2p}$ of out-degree $k$ up to isomorphism for each $k$.
Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.
Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $Γ_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $Γ_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $Γ_n$ is equal to $\mathrm{Aut}(Γ_n)=R(S_n)\rtimes \langle\mathrm{Inn}(ϕ)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $ϕ=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(ϕ)$ is the inner isomorphism of $S_n$ induced by $ϕ$.
Let $\mathcal{G}(4,2)$ be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, $\mathcal{G}(4,2,-1)$ (resp. $\mathcal{G}(4,2,0)$) the set of graphs belonging to $\mathcal{G}(4,2)$ with $-1$ (resp. $0$) as an eigenvalue, and $\mathcal{G}(4,\geq -1)$ the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than $-1$. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in $\mathcal{G}(4,2,-1)$. As a by-product of this work, we characterize all the graphs belonging to $\mathcal{G}(4,\geq-1)$ and $\mathcal{G}(4,2,0)$, respectively, and show that all these graphs are determined by their spectra.
In this paper, we completely characterize the graphs with third largest distance eigenvalue at most $-1$ and smallest distance eigenvalue at least $-3$. In particular, we determine all graphs whose distance matrices have exactly two eigenvalues (counting multiplicity) different from $-1$ and $-3$. It turns out that such graphs consist of three infinite classes, and all of them are determined by their distance spectra. We also show that the friendship graph is determined by its distance spectrum.
In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over $D_n$ in terms of the Boolean algebra of $\langle a\rangle$, from which we find infinite classes of integral Cayley graphs over $D_n$. In particular, we completely determine all integral Cayley graphs over the dihedral group $D_p$ for a prime $p$.