Mean first passage time for distance-biregular graphs

Postprint (published version

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

Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4

The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$

On strongly closed subgraphs of highly regular graphs

AbstractA geodetically closed induced subgraph Δ of a graph Γ is defined to be strongly closed if Γi(α) ∩ Γ1(β) stays in Δ for every i and α, β ϵ Δ with ∂(α, β) = i. We study the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs

Distance-regular graph with large a1 or c2

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.Comment: We submited this manuscript to JCT

Characterising vertex-star transitive and edge-star transitive graphs

Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k, L)-complex. The two conditions are symmetry properties of the graph, namely vertex-star transitivity and edge-star transitivity. In this paper we investigate vertex- and edge-star transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both vertex-star transitive and edge-star transitive

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

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