On conjectures of network distance measures by using graph spectra

TL;DR

This paper addresses three open conjectures about graph spectrum-based network distance measures by constructing counterexamples and employing computer search, thus advancing understanding of spectral graph metrics.

Contribution

It resolves three conjectures on spectral network distance measures through counterexamples and computational methods, clarifying their limitations.

Findings

Counterexamples disprove the conjectures

Computer search aids in identifying spectral measure properties

Clarifies the boundaries of spectral network distance measures

Abstract

In this note we resolve three conjectures from [M. Dehmer, S. Pickl, Y. Shi, G. Yu, \emph{New inequalities for network distance measures by using graph spectra}, Discrete Appl. Math. 252 (2019), 17--27] on the comparison of distance measures based on the graph spectra, by constructing families of counterexamples and using computer search.