Strongly walk-regular graphs

  • Authors:

  • E. R. Van Dam;G. R. Omidi

  • Affiliations:

  • Tilburg University, Dept. Econometrics and Operations Research, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands;Dept. Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehr ...

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an @?1 such that the number of walks of length @? 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 @?-walk-regular for all @?, whereas the graphs from the fourth family are @?-walk-regular for every odd @?. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly @?-walk-regular for even @?. We will characterize the case that regular four-eigenvalue graphs are strongly @?-walk-regular for every odd @?, 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 @?-walk-regular for at most one @?. 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 @?-walk-regular for only one particular @? different from 3.