12 research outputs found
Weakly distance-regular circulants, I
We classify certain non-symmetric commutative association schemes. As an
application, we determine all the weakly distance-regular circulants of one
type of arcs by using Schur rings. We also give the classification of primitive
weakly distance-regular circulants.Comment: 28 page
Distance-regular Cayley graphs with small valency
We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most , the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency , and the Cayley graphs among all
distance-regular graphs with girth and valency or . We obtain that
the incidence graphs of Desarguesian affine planes minus a parallel class of
lines are Cayley graphs. We show that the incidence graphs of the known
generalized hexagons are not Cayley graphs, and neither are some other
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.Comment: 19 pages, 4 table
Graphs with k-Uniform Edge Betweenness Centrality
The edge betweenness centrality of an edge is defined as the ratio of shortest paths between all pairs of vertices passing through that edge. A graph is said to have k-uniform edge betweenness centrality, if there are k different edge betweenness centrality values. In this thesis we investigate graphs that have k-uniform edge betweenness centrality where k=2 or 3, and precisely determine the edge betweenness centrality values for various families of graphs
On the independence ratio of distance graphs
A distance graph is an undirected graph on the integers where two integers
are adjacent if their difference is in a prescribed distance set. The
independence ratio of a distance graph is the maximum density of an
independent set in . Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM
J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is
equal to the inverse of the fractional chromatic number, thus relating the
concept to the well studied question of finding the chromatic number of
distance graphs.
We prove that the independence ratio of a distance graph is achieved by a
periodic set, and we present a framework for discharging arguments to
demonstrate upper bounds on the independence ratio. With these tools, we
determine the exact independence ratio for several infinite families of
distance sets of size three, determine asymptotic values for others, and
present several conjectures.Comment: 39 pages, 12 figures, 6 table
Distance-regular Cayley graphs over (pseudo-) semi-dihedral groups
Distance-regular graphs are a class of regualr graphs with pretty
combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the
problem of charaterizing distance-regular Cayley graphs, which can be viewed as
a natural extension of the problem of characterizing strongly-regular Cayley
graphs (or equivalently, regular partial difference sets). In this paper, we
provide a partial characterization for distance-regular Cayley graphs over
semi-dihedral groups and pseudo-semi-dihedral groups, both of which are
-groups with a cyclic subgroup of index .Comment: 21 page