86 research outputs found
Four-class Skew-symmetric Association Schemes
An association scheme is called skew-symmetric if it has no symmetric
adjacency relations other than the diagonal one. In this paper, we study
4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of
skew-symmetric amorphous association schemes, submitted for publication], we
discovered that their character tables fall into three types. We now determine
their intersection matrices. We then determine the character tables and
intersection numbers for 4-class skew-symmetric pseudocyclic association
schemes, the only known examples of which are cyclotomic schemes. As a result,
we answer a question raised by S. Y. Song [Commutative association schemes
whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996].
We characterize and classify 4-class imprimitive skew-symmetric association
schemes. We also prove that no 2-class Johnson scheme can admit a 4-class
skew-symmetric fission scheme. Based on three types of character tables above,
a short list of feasible parameters is generated.Comment: 12 page
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Descriptive complexity of graph spectra.
Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic are cospectral, and this is not the case with , nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.OZ was supported by CONACyT-Mexico Grant 384665, SS was supported by EPSRC and The Royal Society
Existential Closure in Line Graphs
A graph is -existentially closed if, for all disjoint sets of vertices
and with , there is a vertex not in adjacent
to each vertex of and to no vertex of .
In this paper, we investigate -existentially closed line graphs. In
particular, we present necessary conditions for the existence of such graphs as
well as constructions for finding infinite families of such graphs. We also
prove that there are exactly two -existentially closed planar line graphs.
We then consider the existential closure of the line graphs of hypergraphs and
present constructions for -existentially closed line graphs of hypergraphs.Comment: 13 pages, 2 figure
Average mixing of continuous quantum walks
If is a graph with adjacency matrix , then we define to be the
operator . The Schur (or entrywise) product is a
doubly stochastic matrix and, because of work related to quantum computing, we
are concerned the \textsl{average mixing matrix}. This can be defined as the
limit of C^{-1} \int_0^C H(t)\circ H(-t)\dt as . We establish
some of the basic properties of this matrix, showing that it is positive
semidefinite and that its entries are always rational. We find that for paths
and cycles this matrix takes on a surprisingly simple form, thus for the path
it is a linear combination of , (the all-ones matrix), and a permutation
matrix.Comment: 20 pages, minor fixes, added section on discrete walks; fixed typo
Computational and Theoretical Aspects of \u3cem\u3eN\u3c/em\u3e-E.C. Graphs
We consider graphs with the n-existentially closed adjacency property. For a positive integer n, a graph is n-existentially closed (or n-e.c.) if for all disjoint sets of vertices A and B with \A∪ B\ = n (one of A or B can be empty), there is a vertex 2 not in A∪B joined to each vertex of A and no vertex of B. Although the n-e.c. property is straightforward to define, it is not obvious from the definition that graphs with the property exist. In 1963, Erdos and Rényi gave a non-explicit, randomized construction of such graphs. Until recently, only a few explicit families of n-e.c. graphs were known such as Paley graphs. Furthermore, n-e.c. graphs of minimum order have received much attention due to Erdos’ conjecture 011 the asymptotic order of these graphs. The exact minimum orders are only known for n = 1 and n = 2.
We provide a survey of properties and examples of n-e.c. graphs. Using a computer search, a new example of a 3-e.c. graph of order 30 is presented. Previously, no known 3-e.c. graph was known to exist of that order. We give a new randomized construction of n-e.c. vertex-transitive graphs, exploiting Cayley graphs. The construction uses only elementary probability and group theory
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