Let us consider a connected graph G with diameter D. For a given integer k between 0 and D, we say that G is k-walk-regular if the number of walks of length between vertices u, v only depends on the distance between u and v, provided that such a distance does not exceed k. Thus, in particular, a 0-walk-regular graph is the same as a walk-regular graph, where the number of cycles of length roote...

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G . Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain...

A lower bound is given for the harmonic mean of the growth in a finite undirected graph 1 in terms of the eigenvalues of the Laplacian of 1. For a connected graph, this bound is tight if and only if the graph is distance-regular. Bounds on the diameter of a ``sphere-regular'' graph follow. Finally, a lower bound is given for the growth in an infinite undirected graph of bounded degree in terms ...

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 3-regular 4-ordered graphs other than K4 and K3,3. In 2008, Meszaros solve the question by proving the Petersen graph and the Heawood graph are 3-regular 4-ordered gr...

We are interested in (ε)-regular bipartite graphs which are the central objects in the regularity lemma of Szemerédi for sparse graphs. A bipartite graph G = (A]B,E) with density p = |E|/(|A||B|) is (ε)-regular if for all sets A′ ⊆ A and B′ ⊆ B of size |A′| ≥ ε|A| and |B ′| ≥ ε|B|, it holds that |eG(A, B′)/(|A′||B′|) − p| ≤ εp. In this paper we prove a characterization for (ε)-regularity. That ...

A P -decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an -regular graph to admit a P -decomposition, for odd . Let g, and m be positive integers wi...

A b-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The b-chromatic number of a graph G, denoted by b(G), is the maximum integer k such that G admits a b-coloring with k colors. El Sahili and Kouider conjectured that b(G) = d + 1 for d-regular graph with girth 5, d ≥ 4. In this paper, we p...

In 1891, Peterson [6] proved that every 3-regular bridge-less graph has a perfect matching. It is well known that the dual of a triangular mesh on a compact manifold is a 3-regular graph. M. Gopi and D. Eppstein [4] use Peterson’s theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [3] elaborate on the crea...

A graph is r-regular if all its vertices have the same degree r. A random r-regular graph with n vertices is a graph sampled uniformly at random from the set of all r-regular graphs on n vertices. There are various sampling methods for this. In this report we survey one of them: The Configuration Model. We then survey the cover time of a random walk on such a graph, which roughly speaking, is t...