On distance-biregular graphs

• Main Author: Mateo, Rolando A.
• Format: text
• Published: Animo Repository 2002
• Subjects: Graph theory; Graphic methods; Combinatorial analysis; Applied Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/2832
• Institution: De La Salle University

This study is an exposition of Kazumasa Nomura's article entitled Intersection Diagrams of Distance-Biregular Graphs published in the Journal of Combinatorial Theory, Series B, Volume 50, No. 2, December 1990. The paper aims to:1. Present elementary properties of distance-biregular graphs2. Provide and alternative proof to the following theorem If a distance graph G has 2 valent vertices, G is the subdivision graph of a (K,g)-graph.3. Characterize distance-biregular graphs of girth g-=0 (mod 4)Basic concepts on distance biregular graphs were discussed. The highlight of the paper was the presentation of results using intersection diagram. Among the main results discussed were:1. If a distance graph G has 2-valent vertices, G is isomorphic to the subdivision graph of a (k,g)-graph.2. Let G be a distance-biregular graph with girth g=0(mod 4). Let s + 1 and t + 1 be the valencies of G and assume that s and t are relatively prime. Then G is isomorphic to a generalized polygon.3. Let G be a distance-biregular graph of girth g=0(mod4). Assume G has the valency s+1 and 3. Then one of the following two cases occurs:(i) G is a generalized polygon (ii) s=2h - 2 for some h less than or equal to g/2 and d(G) less than or equal to g/2 + g/h This study concentrated on the use of intersection diagram to prove a result earlier obtained by Mohar and Shawe-Taylor. Moreover, discussion was limited to graphs with girth g = 0 (mod 4) and containing 2-valent vertices. It is recommended that the intersection diagrams be used to study other families of distance-biregular graphs. In particular,