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arXiv · 2608.15992

Two conjectures on graphs and their edge-path matrices

Metrose Metsidik·Xian'an Jin
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Abstract

The edge-path matrix is a square matrix where each off-diagonal entry records the maximum number of edge-disjoint paths between the corresponding pair of vertices. Akbari et al. [On edge-path eigenvalues of graphs, Linear Multilinear Algebra 70 (2022) 2998-3008] proposed two conjectures: Conjecture 1 relates the edge-path matrix to an upper bound on the number of edges in the graph, while Conjecture 2 asserts that a graph is Eulerian if and only if all entries of its edge-path matrix are even. In this paper, we prove the two conjectures.

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Metrose Metsidik, Xian'an Jin. 2026-08-17. Two conjectures on graphs and their edge-path matrices. https://arxiv.org/abs/2608.15992

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