Regularity and transitivity in graphs

Graphs with high regularity and transitivity conditions are studied. The first graphs considered are graphs where each vertex has an intersection array (possibly differing from that of other vertices). These graphs are called distance-regularised and are shown to be distance-regular or bipartite with each bipartition having the same intersection array. The latter graphs are called distance-biregular. This leads to the study of distance-biregular graphs. The derived graphs of a distance-biregular graph are shown to be distance-regular and the notion of feasibility for a distance-regular graph is extended to the biregular case. The study of the intersection arrays of distance-biregular graphs is concluded with a bound on the diameter in terms of the girth and valencies. Special classes of distance-biregular graphs are also studied. Distance-biregular graphs with 2-valent vertices are shown to be the subdivision graphs of cages. Distance-biregular graphs with one derived graph complete and the other strongly-regular are characterised according to the minimum eigenvalue of the strongly-regular graph. Distance-biregular graphs with prescribed derived graph are classified in cases where the derived graph is from some classes of classical distance-regular graphs. A graph theoretic proof of part of the Praeger, Saxl and Yokoyama theorem is given. Finally imprimitivity in distance-biregular graphs is studied and the Praeger, Saxl and Yokoyama theorem is used to show that primitive non-regular distance-bitransitive graphs have almost simple automorphism groups. Many examples of distance-biregular and distance-bitransitive graphs are given.<p

Locally ss-distance transitive graphs

We give a unified approach to analysing, for each positive integer $s$, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally $s$-arc transitive graphs of diameter at least $s$. A graph is in the class if it is connected and if, for each vertex $v$, the subgroup of automorphisms fixing $v$ acts transitively on the set of vertices at distance $i$ from $v$, for each $i$ from 1 to $s$. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for $s\geq 2$, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report

Distance-Biregular Graphs and Orthogonal Polynomials

This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound

Automorphism groups of primitive distance-bitransitive graphs are almost simple

We apply a result of Praeger, Saxl and Yokoyama [4] concerning the automorphism groups of primitive distance-transitive graphs to primitive distance-bitransitive graphs. Imprimitive distance-bitransitive graphs are discussed in Section 2. One of the cases of the Praeger, Saxl and Yokayama Theorem is considered in Section 3 and the following main result is proved in Section 4