Median eigenvalues of sparse subcubic graphs

For an n-vertex graph G, let λ1(G)≥λ2(G)≥…≥λn(G) be the sequence of eigenvalues of its adjacency matrix. The HL-index of an n-vertex graph G, denoted by R(G), is defined as R(G)=max⁡{|λ⌊n+12⌋(G)|,|λ⌈n+12⌉(G)|}. Mohar proved in 2015 that every subcubic graph G satisfies that R(G)≤2 and conjectured th...

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Bibliographic Details
Published inDiscrete mathematics Vol. 348; no. 12; p. 114611
Main Authors Chen, Zhengbo, Wang, Zhouningxin, Zhang, Xiao-Dong
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.12.2025
Subjects
HL-index
Median eigenvalues
Planar graphs
Subcubic graph
Subcubic graph
HL-index
Median eigenvalues
Planar graphs
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Summary:For an n-vertex graph G, let λ1(G)≥λ2(G)≥…≥λn(G) be the sequence of eigenvalues of its adjacency matrix. The HL-index of an n-vertex graph G, denoted by R(G), is defined as R(G)=max⁡{|λ⌊n+12⌋(G)|,|λ⌈n+12⌉(G)|}. Mohar proved in 2015 that every subcubic graph G satisfies that R(G)≤2 and conjectured that R(G)≤1 when restricted to planar graphs. Bipartite subcubic graphs and K4-minor-free subcubic graphs have been verified to satisfy this conjecture. In this paper, we prove that two classes of sparse subcubic graphs G satisfy that R(G)≤1: Subcubic graphs with maximum average degree smaller than 4417 and subcubic planar graphs of girth at least 8.
ISSN:0012-365X
DOI:10.1016/j.disc.2025.114611