Integral equienergetic non-isospectral unitary Cayley graphs

On regular graphs equienergetic with their complements

We give necessary and sufficient conditions on the parameters of a regular graph $Γ$ (with or without loops) such that $E(Γ)=E(\overline Γ)$. We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ($Γ= K_3 \square K_2$ or $Q_3$). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs $Cr(n)$ or $C_4$. Next, for the family of strongly regular …

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs $Γ(k,q)$, generalized Paley sum graphs $Γ^+(k,q)$, and their corresponding complements $\bar Γ(k,q)$ and $\bar Γ^+(k,q)$, for $k=3,4$. Denote by $Γ= Γ^*(k,q)$ either $Γ(k,q)$ or $Γ^+(k,q)$. We compute the spectra of $Γ(3,q)$ and $Γ(4,q)$ and from them we obtain the spectra of $Γ^+(3,q)$ and $Γ^+(4,q)$ also. Then we show that, in the non-semiprimitive case, the spectrum of $Γ(3,p^{3\ell})$ and $Γ(4,p^{4\ell})$ with $p$ prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs $Γ(3,p)$ and $Γ(4,p)$ for any $\ell \…

有限单群的等谱Cayley图

Isospectral Cayley graphs of some finite simple groups

利用谱分析方法，对每个$d \geq 5$（$d \neq 6$）和素数幂$q > 2$，构造了有限单群$\operatorname{PSL}_d({\mathbb F}_q)$的等谱非同构Cayley图.

具有偶谱和奇谱的等谱Cayley图

Isospectral Cayley graphs with even and odd spectrum

本文研究了镜像双Cayley图的谱性质，发现当T属于特定集合时，图的谱为偶或奇，从而构造了偶谱和奇谱的等谱Cayley图对。

同构与非同构的等谱循环图

Isomorphic and Nonisomorphic, Isospectral Circulant Graphs

本文给出了任意阶循环群的Cayley图可由其邻接矩阵的特征值完全确定直至同构的新判据。其次，给出了具有相同邻接矩阵特征值表的循环群的非同构Cayley图对的新构造。

Expansion in Cayley graphs, Cayley sum graphs and their twists

The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and Vishne proved that there are isospectral non-isomorphic Cayley graphs of $\mathrm{PSL}_d(\mathbb F_q)$ for every $d\geq 5$ ($d \neq 6$) and prime power $q> 2$. In this article, we focus on three variants of Cayley graphs, viz., the Cayley sum graphs, the twisted Cayley graphs, and the twisted Cayley sum graphs. We prove the existence of non-isomorphic expande…