Invariant adjacency matrices of configuration graphs

Abreu, M.; Funk, Martin; Labbate, Domenico; Napolitano, Vito

doi:10.1016/j.laa.2012.05.029

Cited by 1 publication

(1 citation statement)

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“…At this point it seems appropriate to remark that Abreu, Funk, Labbate, and Napolitano [1] recently generalized the (strongly regular) Moore graphs with girth 5 by considering the equation ((k − 1)I + J − A 2 ) m = A. This is however a quite different equation in terms of A, I, and J than the one we are considering.…”

Section: The Definition and Some Basic Observationsmentioning

confidence: 88%

Strongly walk-regular graphs

Dam

Omidi

2013

Journal of Combinatorial Theory, Series A

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We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-walk-regular for all $\ell$, whereas the graphs from the fourth family are $\ell$-walk-regular for every odd $\ell$. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly $\ell$-walk-regular for even $\ell$. We will characterize the case that regular four-eigenvalue graphs are strongly $\ell$-walk-regular for every odd $\ell$, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly $\ell$-walk-regular for at most one $\ell$. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly $\ell$-walk-regular for only one particular $\ell$ different from 3

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Section: The Definition and Some Basic Observationsmentioning

confidence: 88%