The Laplacian eigenvalue 2 of bicyclic graphs

2019/09/14 by Doost Ali Mojdeh, Mojdeh, Doost Ali, Mohammad Habibi +3
Chemistry · Computer Science · Mathematics · #15A18 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.1909.06578

openalex publication_date 2019/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If G is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G₁ and G₂ is a graph G=G₁\odot G₂ with V(G)=V(G₁)∪ V(G₂) and E(G)= E(G₁)∪ E(G₂)∪ \e=uv\ where u∈ V(G₁) and v∈ V(G₂). In this paper, we consider the eigenvector of unicycle graphs. We study the relationship between the Laplacian eigenvalue 2 of unicyclic graphs G₁ and G₂; and bicyclic graphs G=G₁\odot G₂. We also characterize the broken sun graphs and the one edge connection of two broken sun graphs by their Laplacian eigenvalue 2.

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