Six signed Petersen graphs, and their automorphisms

Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.Comment: 39 pp., 7 fi

A note on Nordhaus-Gaddum-type inequaliies for the automorphic H-chromatic index of graphs

The automorphic H-chromatic index of a graph G is the minimum integer m for which G has a proper edge-coloring with m colors which is preserved by a given automorphism H of G. We consider the sum and the product of the automorphic H-chromatic index of a graph and its complement. We prove upper and lower bounds in terms of the order of the graph when H is chosen to be either a cyclic group of prime order or a group of order four

Automorphic chromatic index of generalized Petersen graphs

The automorphic A-chromatic index of agraph G is the minimum integer m for which G has aproper edge-coloring with m colors which is preserved by a givensubgroup A of the full automorphism group of G. We computethe automorphic A-chromatic index of each generalized Petersengraph when A is the full automorphism group

Finding Edge and Vertex Induced Cycles within Circulants.

Let H be a graph. G is a subgraph of H if V (G) ⊆ V (H) and E(G) ⊆ E(H). The subgraphs of H can be used to determine whether H is planar, a line graph, and to give information about the chromatic number. In a recent work by Beeler and Jamison [3], it was shown that it is difficult to obtain an automorphic decomposition of a triangle-free graph. As many of their examples involve circulant graphs, it is of particular interest to find triangle-free subgraphs within circulants. As a cycle with at least four vertices is a canonical example of a triangle-free subgraph, we concentrate our efforts on these. In this thesis, we will state necessary and sufficient conditions for the existence of edge induced and vertex induced cycles within circulants

Combinatorial aspects of Hecke algebra characters

Iwahori-Hecke algebras are deformations of Coxeter group algebras. Their origins lie in the theory of automorphic forms but they arise in the representation theory of Coxeter groups and Lie algebras and in quantum group theory. The Kazhdan-Lusztig bases of these algebras, originally introduced in the late 1970s in connection with representation-theoretic concerns, has turned out to have deep connections to Schubert varieties, intersection cohomology, and related topics.Matrix immanants were originally introduced by Littlewood as a generalization of determinants and permanants. They remained obscure until the 1980s when their connections to symmetric function and representation theory as well as their surprising algebraic and combinatorial properties came to light. In particular, it was discovered that they have a fruitful connection to the theory of total positivity. More recently, a theory of quantum immamants was developed, providing a bridge to the quantum group theory.In this paper we develop the theory of certain planar networks, which provide a unified combinatorial setting for these fields of study. In particular, we use these networks to evaluate certain characters of the symmetric group algebra. We give new combinatorial interpretations of the quantum induced sign and trivial characters of the type A Iwahori-Hecke algebras