ABSTRACT

Let G be a graph with vertex set $V\left(G\right)=\{v_1,\dots ,v_n\}$ and $EP\left(G\right)$ be an $n\times n$ matrix whose $(i,j)$-entry is the maximum number of internally edge-disjoint paths between $v_i$ and $v_j$, if $i\ne j$, and zero otherwise. Also, define $\overline{EP}\left(G\right)=EP\left(G\right)+D$, where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing $v_i$, whose $EP\left(G\right)$ is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix $\overline{EP}\left(G\right)$ for an arbitrary bicyclic graph G.

中文翻译：

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关于图的边路径特征值

摘要

设G是一个有顶点集的图$五\left(G\right)=\{v_1,\dots ,v_n\}$和$乙磷\left(G\right)$豆$n\times n$矩阵$(\mathrm{一世},j)$-entry 是内部边缘不相交路径的最大数量$v_{\mathrm{一世}}$和$v_j$， 如果$\mathrm{一世}\ne j$，否则为零。另外，定义$\overline{乙磷}\left(G\right)=乙磷\left(G\right)+D$，其中D是一个对角矩阵，其第i个对角元素是包含的边不相交循环的数量$v_{\mathrm{一世}}$，谁的$乙磷\left(G\right)$是J − I的倍数。在其他结果中，我们确定了矩阵的光谱和能量$\overline{乙磷}\left(G\right)$对于任意双环图G。