On edge-path eigenvalues of graphs

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.

Keywords:

• Eigenvalue
• edge-path energy
• edge-connectivity

Acknowledgments

The authors would like to thank an anonymous referee for comments and suggestions. The research of the first author was partly funded by Iran National Science Foundation(INSF) under the contract No. 96004167.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the first author was partly funded by Iran National Science Foundation(INSF) under the contract No. 96004167.