18 research outputs found
Brick assignments and homogeneously almost self-complementary graphs
AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the âcocktail party graphsâ K2nânK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices
On finite edge-primitive and edge-quasiprimitive graphs
Many famous graphs are edge-primitive, for example, the Heawood graph, the
Tutte--Coxeter graph and the Higman--Sims graph. In this paper we
systematically analyse edge-primitive and edge-quasiprimitive graphs via the
O'Nan--Scott Theorem to determine the possible edge and vertex actions of such
graphs. Many interesting examples are given and we also determine all
-edge-primitive graphs for an almost simple group with socle .Comment: 30 pages To appear in Journal of Combinatorial Theory Series
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Contributions at the Interface Between Algebra and Graph Theory
In this thesis, we make some contributions at the interface between algebra and graph theory.
In Chapter 1, we give an overview of the topics and also the definitions and preliminaries.
In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of
lacunary polynomials and a bound on the so-called domination number of a graph.
In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and
Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way
to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero.
In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of
these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the
well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning
trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we
do not use zeta functions in our arguments.
In Chapter 5, we present the conclusion and also some possible projects
A new family of posets generalizing the weak order on some Coxeter groups
We construct a poset from a simple acyclic digraph together with a valuation
on its vertices, and we compute the values of its M\"obius function. We show
that the weak order on Coxeter groups of type A, B, affine A, and the flag weak
order on the wreath product introduced by Adin,
Brenti and Roichman, are special instances of our construction. We conclude by
associating a quasi-symmetric function to each element of these posets. In the
and cases, this function coincides respectively with the
classical Stanley symmetric function, and with Lam's affine generalization