论文标题
在常规图上，其补语
On regular graphs equienergetic with their complements
论文作者
论文摘要
我们在常规图$γ$（有或没有循环的）的参数上提供了必要的条件，使得$ e（γ）= e（\ edimenlineγ）$。我们研究了具有单个环（5个非镜面对）连接的立方图的同构和连接的无环的积分立方图（$γ= k_3 \ k_3 \ square k_2 $或$ q_3 $）的互补分类图的互补分类图。然后，我们证明，为了补充，唯一的双分部分均值图形和非镜头及其补充是冠状图$ cr（n）$或$ c_4 $。接下来，对于强烈规则图$γ$的家族，我们表征所有可能的参数$ srg（n，k，e，d）$，使得$ e（γ）= e（\ overlineγ）$。此外，使用此功能，我们证明，当它是会议图，或者它是伪拉丁方形图（即具有$ oa $ parameters）时，我们就证明了强烈规则的图表与其补充相等。我们还表征了类型$ \ MATHCAL {C}（2）$，$ \ MATHCAL {C}（3）$和$ \ MATHCAL {C}（5）$在Cameron's HierArchy中（情况$ \ MATHCAL {C}（c}（1）$ nys $ nys $ nys $ nys $，最后，我们考虑戒指上的统一Cayley图$ g_r = x（r，r^*）$。我们表明，如果$ r $是具有偶数局部因素的有限artinian戒指，那么$ g_r $是且仅当$ r = \ mathbb {f} _q \ times \ times \ times \ times \ mathbb {f} _ {q'} $的产品是2个有限的领域的产物。
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 graphs $Γ$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(Γ) = E(\overline Γ)$. Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has $OA$ parameters). We also characterize all complementary equienergetic pairs of graphs of type $\mathcal{C}(2)$, $\mathcal{C}(3)$ and $\mathcal{C}(5)$ in Cameron's hierarchy (the cases $\mathcal{C}(1)$ and $\mathcal{C}(4)$ are still open). Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We show that if $R$ is a finite Artinian ring with an even number of local factors, then $G_R$ is complementary equienergetic if and only if $R=\mathbb{F}_q \times \mathbb{F}_{q'}$ is the product of 2 finite fields.
