两个图的邻域冠的谱
Spectra of the neighbourhood corona of two graphs
摘要：
给定单图$G_1$和$G_2$，$G_1$和$G_2$的邻域冠，记作$G_1\star G_2$，是通过取$G_1$的一个副本和$G_2$的$|V(G_1)|$副本，并将$G_1$的$i$第个顶点的邻居连接到$G_2$的$i$第个副本中的每个顶点而获得的图。本文根据$G_1$和$G_2$的相应谱，确定了任意$G_1$和$G_2$的$G_1 \star G_2$的邻接谱，以及正则$G_1$和任意$G_2$的$G_1\star G_2$的拉普拉斯谱和无符号拉普拉斯谱.关于邻接和无号Laplacian谱的结果使我们能够构造新的邻接共谱和无号Laplacian共谱图对。作为这些结果在Laplacian谱上的应用，我们利用邻域冠构造了一类新的扩张图族。
英文摘要：
Given simple graphs $G_1$ and $G_2$, the neighbourhood corona of $G_1$ and $G_2$, denoted $G_1\star G_2$, is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we determine the adjacency spectrum of $G_1 \star G_2$ for arbitrary $G_1$ and $G_2$, and the Laplacian spectrum and signless Laplacian spectrum of $G_1\star G_2$ for regular $G_1$ and arbitrary $G_2$, in terms of the corresponding spectrum of $G_1$ and $G_2$. The results on the adjacency and signless Laplacian spectra enable us to construct new pairs of adjacency cospectral and signless Laplacian cospectral graphs. As applications of the results on the Laplacian spectra, we give constructions of new families of expander graphs from known ones by using neighbourhood coronae.
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