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Strongly walk-regular graphs
We study a generalization of strongly regular graphs. We call a graph
strongly walk-regular if there is an 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
Dynamic Chromatic Number of Regular Graphs
A dynamic coloring of a graph is a proper coloring such that for every
vertex of degree at least 2, the neighbors of receive at least
2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}.
PhD thesis, West Virginia University, 2001.] that if is a -regular
graph, then . In this paper, we prove that if is a
-regular graph with , then . It confirms the conjecture for all regular graph with
diameter at most 2 and . In fact, it shows that
provided that has diameter at most 2 and
. Moreover, we show that for any -regular graph ,
. Also, we show that for any there exists a
regular graph whose chromatic number is and .
This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A.
Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number
and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In
press].Comment: 8 page
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