40 research outputs found
Quasi m-Cayley circulants
A graph ▫▫ is called a quasi ▫▫-Cayley graph on a group ▫▫ if there exists a vertex ▫▫ and a subgroup ▫▫ of the vertex stabilizer ▫▫ of the vertex ▫▫ in the full automorphism group ▫▫ of ▫▫, such that ▫▫ acts semiregularly on ▫▫ with ▫▫ orbits. If the vertex ▫▫ is adjacent to only one orbit of ▫▫ on ▫▫, then ▫▫ is called a strongly quasi ▫▫-Cayley graph on ▫▫ .In this paper complete classifications of quasi 2-Cayley, quasi 3-Cayley and strongly quasi 4-Cayley connected circulants are given
Some conditions implying stability of graphs
A graph is said to be unstable if the direct product (also
called the canonical double cover of ) has automorphisms that do not come
from automorphisms of its factors and . It is non-trivially unstable
if it is unstable, connected, non-bipartite, and distinct vertices have
distinct sets of neighbours. In this paper, we prove two sufficient conditions
for stability of graphs in which every edge lies on a triangle, revising an
incorrect claim of Surowski and filling in some gaps in the proof of another
one. We also consider triangle-free graphs, and prove that there are no
non-trivially unstable triangle-free graphs of diameter 2. An interesting
construction of non-trivially unstable graphs is given and several open
problems are posed.Comment: 13 page
Stability of Cayley graphs and Schur rings
A graph is said to be unstable if for the direct product , is not isomorphic to . In this paper we show that a connected and non-bipartite Cayley
graph is unstable if and only if the set belongs to
a Schur ring over the group having certain properties.
The Schur rings with these properties are characterized if is an abelian
group of odd order or a cyclic group of twice odd order. As an application, a
short proof is given for the result of Witte Morris stating that every
connected unstable Cayley graph on an abelian group of odd order has twins
(Electron.~J.~Combin, 2021). As another application, sufficient and necessary
conditions are given for a connected and non-bipartite circulant graph of order
to be unstable, where is an odd prime and
On Almost Well-Covered Graphs of Girth at Least 6
We consider a relaxation of the concept of well-covered graphs, which are
graphs with all maximal independent sets of the same size. The extent to which
a graph fails to be well-covered can be measured by its independence gap,
defined as the difference between the maximum and minimum sizes of a maximal
independent set in . While the well-covered graphs are exactly the graphs of
independence gap zero, we investigate in this paper graphs of independence gap
one, which we also call almost well-covered graphs. Previous works due to
Finbow et al. (1994) and Barbosa et al. (2013) have implications for the
structure of almost well-covered graphs of girth at least for . We focus on almost well-covered graphs of girth at least . We show
that every graph in this class has at most two vertices each of which is
adjacent to exactly leaves. We give efficiently testable characterizations
of almost well-covered graphs of girth at least having exactly one or
exactly two such vertices. Building on these results, we develop a
polynomial-time recognition algorithm of almost well-covered
-free graphs