Tricyclic graphs for which the second largest distance eigenvalue less than -(1)/(2)

2025/09/16 by Yang, Kexin, Ligong Wang, Wang, Ligong
Chemistry · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.2509.12640

openalex publication_date 2025/09/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

Let G be a simple connected graph with vertex set V(G)=\v₁, v₂, …, v_n\. The distance d_G(v_i,v_j) between two vertices v_i and v_j of G is the length of a shortest path between v_i and v_j. The distance matrix of G is defined as D(G)=(d_G(v_i,v_j))_{n× n}. The second largest distance eigenvalue of \( G \) is the second largest eigenvalues of D(G). Guo and Zhou [Discrete Math. 347(2024), 114082] proved that any connected graph with the second largest distance eigenvalue less than -(1)/(2) is chordal, and characterize all bicyclic graphs and split graphs with the second largest distance eigenvalue less than -(1)/(2). Based on this, we characterize all tricyclic graphs with the second largest distance eigenvalue less than -(1)/(2).

Citations

• On the second largest distance eigenvalue of a graph

Related

• A graph for which the second largest distance eigenvalue is less than (-3+√(5))/(2) is chordal
• The least distance eigenvalue of the complements of graphs of diameter greater than three
• The distance-regular graphs such that all of its second largest local eigenvalues are at most one
• Graphs with at most three distance eigenvalues different from -1 and -2
• Graphs with three distinct distance eigenvalues