TY - JOUR
T1 - A Relationship between the Second Largest Eigenvalue and Local Valency of an Edge-regular Graph
AU - Park, Jongyook
N1 - Publisher Copyright:
© 2021. Kyungpook Mathematical Journal
PY - 2021/9
Y1 - 2021/9
N2 - For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that (Formula Presented) if D = 3 and (Formula Presented) if D ≥ 4, where λ = a1. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare (Formula Presented) with the local valency λ to find a relationship between the second largest eigen- value and the local valency. For an edge-regular graph with diameter 3, we look at the number (Formula Presented), where (Formula Presented), and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.
AB - For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that (Formula Presented) if D = 3 and (Formula Presented) if D ≥ 4, where λ = a1. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare (Formula Presented) with the local valency λ to find a relationship between the second largest eigen- value and the local valency. For an edge-regular graph with diameter 3, we look at the number (Formula Presented), where (Formula Presented), and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.
KW - distance-regular graphs
KW - edge-regular graphs
KW - local valency
KW - second largest eigen-values
UR - http://www.scopus.com/inward/record.url?scp=85117325948&partnerID=8YFLogxK
U2 - 10.5666/KMJ.2021.61.3.671
DO - 10.5666/KMJ.2021.61.3.671
M3 - Article
AN - SCOPUS:85117325948
SN - 1225-6951
VL - 61
SP - 671
EP - 677
JO - Kyungpook Mathematical Journal
JF - Kyungpook Mathematical Journal
IS - 3
ER -