On graphs with 2 trivial distance ideals

Alfaro, Carlos A.

doi:10.1016/j.laa.2020.03.017

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Theorem 7 [4] a Connected Graph Has Only One Trivial Distanc...1

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Cited by 4 publications

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“…This could be obtained after showing a characterization of graphs having two trivial distance ideals. However, the latter problem seems to be difficult, since there exist infinitely many minimal induced forbidden graphs [3] (most of them are the same needed in the characterization of the well-known Strong Perfect Graph Theorem).…”

Section: Theorem 7 [4] a Connected Graph Has Only One Trivial Distanc...mentioning

confidence: 99%

Enumeration of cospectral and coinvariant graphs

Abiad

Alfaro

2020

Preprint

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We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, we show a new graph characterization using the Smith normal form of the signless distance Laplacian matrix.

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Section: Theorem 7 [4] a Connected Graph Has Only One Trivial Distanc...mentioning

confidence: 99%