线图和全图不同展度的界
Bounds for Different Spreads of Line and Total Graphs
摘要：
本文研究了关于给定图的线的展度和全图的展度的一些结果。特别地，证明了对于一个具有$m > n \geq 4$的$(n,m)$连通图$G$，$G$的展度小于或等于它的线图的展度，其中等式成立当且仅当$G$是正则二部图.对一类非二部非正则图，证明了图的展度不大于无符号拉普拉斯展度的一个充分条件.此外，我们得到了一个上界的传播的线图的$n$顶点的顶点（边）连通度小于或等于一个正整数$k$。结合特征值交错的技巧，我们得到了连通图的全图的Laplacian和无符号Laplacian扩展的下界。此外，对于正则图，给出了它的全图的扩散度的上下界。
英文摘要：
In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to the spread of its line graph, where the equality holds if and only if $G$ is regular bipartite. A sufficient condition for the spread of the graph not be greater than the signless Laplacian spread for a class of non bipartite and non regular graphs is proved. Additionally, we derive an upper bound for the spread of the line graph of graphs on $n$ vertices having a vertex (edge) connectivity less than or equal to a positive integer $k$. Combining techniques of interlacing of eigenvalues, we derive lower bounds for the Laplacian and signless Laplacian spread of the total graph of a connected graph. Moreover, for a regular graph, an upper and lower bound for the spread of its total graph is given.
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