论文标题
广义的Paley图与他们的补充
Generalized Paley graphs equienergetic with their complements
论文作者
论文摘要
我们考虑了广义的Paley图$γ（K，Q）$，广义Paley Sum Graphs $γ^+（K，Q）$及其相应的补充$ \barγ（k，q）$和$ \ barγ^+（k，q）$，$ k = 3,4 $。用$γ=γ^*（k，q）$ $γ（k，q）$或$γ^+（k，q）$。我们计算$γ（3，Q）$和$γ（4，Q）$的光谱，从中我们获得了$γ^+（3，q）$和$γ^+（4，q）$的光谱。然后我们表明，在非亲眼的情况下，$γ（3，p^{3 \ ell}）$和$γ（4，p^{4 \ ell}）$带有$ p $ prime可以递归地获得，在某些arithmetic条件下，从图表$γ（3，p）$γ（p）$γ（p）$γ（4） \ Mathbb {n} $。使用这些图的光谱，我们在$γ^*（k，q）$的光谱上提供了必要和足够的条件，以使$γ^*（k，q）$和$ \barγ^*（k，q）$在$ k = 3,4 $中是均等的。在以前的工作中，我们已经对所有两分的常规图进行了分类$γ_{bip} $和所有强烈的常规图$γ_{srg} $，它们是互补的equienergetic，即。 \barγ_{srg} \} $是图形的均值对。在这里，我们构建了无限的等方面的非同一常规图$ \ {γ，\barγ\} $，它们既不是双分部分也不是非常规则的。
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 $\ell \in \mathbb{N}$, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of $Γ^*(k,q)$ such that $Γ^*(k,q)$ and $\bar Γ^*(k,q)$ are equienergetic for $k=3,4$. In a previous work we have classified all bipartite regular graphs $Γ_{bip}$ and all strongly regular graphs $Γ_{srg}$ which are complementary equienergetic, i.e.\@ $\{Γ_{bip}, \barΓ_{bip}\}$ and $\{Γ_{srg}, \barΓ_{srg}\}$ are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $\{Γ, \bar Γ\}$ which are neither bipartite nor strongly regular.
