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Divisibility conditions for intersection numbers of certain bipartite distance-regular graphs

Alexander Habib and Mark S. MacLean

Vol. 18 (2025), No. 1, 151–164
Abstract

A connected graph is said to be distance-regular whenever given any two vertices x,y at path-length distance h apart, the number of vertices at distance i from x and j from y is a fixed constant (called an intersection number of the graph) that only depends on h,i,j, and not the vertices x,y. The classification of all distance-regular graphs of sufficiently large diameter is an open problem that, at least for now, seems out of reach. An active area of research is the classification of distance-regular graphs satisfying certain additional properties. This paper is motivated by a paper of Miklavič which found divisibility conditions on the intersection numbers of certain bipartite Q-polynomial distance-regular graphs. We generalize his work to show that the same divisibility conditions hold for a larger set of bipartite distance-regular graphs.

Keywords
distance-regular graph, bipartite graph
Mathematical Subject Classification
Primary: 05E30
Milestones
Received: 10 July 2023
Revised: 20 July 2023
Accepted: 27 July 2023
Published: 24 January 2025

Communicated by Joshua Cooper
Authors
Alexander Habib
Department of Mathematics
The Ohio State University
Columbus, OH
United States
Mark S. MacLean
Mathematics Department
Seattle University
Seattle, WA
United States