8,693 research outputs found
Another construction of edge-regular graphs with regular cliques
We exhibit a new construction of edge-regular graphs with regular cliques
that are not strongly regular. The infinite family of graphs resulting from
this construction includes an edge-regular graph with parameters . We
also show that edge-regular graphs with -regular cliques that are not
strongly regular must have at least vertices.Comment: 7 page
Asymptotic Delsarte cliques in distance-regular graphs
We give a new bound on the parameter (number of common neighbors of
a pair of adjacent vertices) in a distance-regular graph , improving and
generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber
(2014). The new bound is one of the ingredients of recent progress on the
complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun,
Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch
(1991) under certain constraints on the parameters. We also give a simplified
proof of the following asymptotic consequence of Metsch's result: if then each edge of belongs to a unique maximal clique of size
asymptotically equal to , and all other cliques have size
. Here denotes the degree and the number of common
neighbors of a pair of vertices at distance 2. We point out that Metsch's
cliques are "asymptotically Delsarte" when , so families
of distance-regular graphs with parameters satisfying are
"asymptotically Delsarte-geometric."Comment: 10 page
The Structure and Properties of Clique Graphs of Regular Graphs
In the following thesis, the structure and properties of G and its clique graph clt (G) are analyzed for graphs G that are non-complete, regular with degree δ , and where every edge of G is contained in a t -clique. In a clique graph clt (G), all cliques of order t of the original graph G become the clique graph’s vertices, and the vertices of the clique graph are adjacent if and only if the corresponding cliques in the original graph have at least 1 vertex in common. This thesis mainly investigates if properties of regular graphs are carried over to clique graphs of regular graphs. In particular, the first question considered is whether the clique graph of a regular graph must also be regular. It is shown that while line graphs, cl2(G), of regular graphs are regular, the degree difference of the clique graph cl3(R) can be arbitrarily large using δ -regular graphs R with δ ≥ 3. Next, the question of whether a clique graph can have a large independent set is considered (independent sets in regular graphs can be composed of half the vertices in the graph at the most). In particular, the relation between the degree difference and the independence number of clt (G) will be analyzed. Lastly, we close with some further questions regarding clique graphs
The smallest strictly Neumaier graph and its generalisations
A regular clique in a regular graph is a clique such that every vertex
outside of the clique is adjacent to the same positive number of vertices
inside the clique. We continue the study of regular cliques in edge-regular
graphs initiated by A. Neumaier in the 1980s and attracting current interest.
We thus define a Neumaier graph to be an non-complete edge-regular graph
containing a regular clique, and a strictly Neumaier graph to be a non-strongly
regular Neumaier graph. We first prove some general results on Neumaier graphs
and their feasible parameter tuples. We then apply these results to determine
the smallest strictly Neumaier graph, which has vertices. Next we find the
parameter tuples for all strictly Neumaier graphs having at most vertices.
Finally, we give two sequences of graphs, each with element a
strictly Neumaier graph containing a -regular clique (where is a
positive integer) and having parameters of an affine polar graph as an
edge-regular graph. This answers questions recently posed by G. Greaves and J.
Koolen
- …