The multiplicity of the Laplacian eigenvalue 2 in some bicyclic graphs

2019/04/28 by Masoumeh Farkhondeh, Mohammad Habibi, Farkhondeh, Masoumeh +5
Mathematics · #05C50 #11C08 #15A18 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C50 #msc:11C08 #msc:15A18

paper · pdf · doi:10.48550/arxiv.1904.12299

10 pages

arxiv created 2020/03/07 · arxiv updated 2020/03/10

Abstract

The Laplacian matrix of a graph G is denoted by L(G)=D(G)-A(G), where D(G)=diag(d(v₁),… , d(v_n)) is a diagonal matrix and A(G) is the adjacency matrix of G. Let G₁ and G₂ be two graphs. A one-edge connection of two graphs G₁ and G₂ is a graph G=G₁\odot_uv 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). We investigate the multiplicity of the Laplacian eigenvalue 2 of G₁\odot_uv G₂, while the unicyclic graphs G₁ and G₂ have 2 among their Laplacian eigenvalues, by using their Laplacian characteristic polynomials. Some structural conditions ensuring the presence of the existence 2 in the G=G₁\odot_uv G₂ where both G₁ and G₂ have 2 as Laplacian eigenvalue, have been investigated, while, here we study the existence Laplacian eigenvalue 2 in G=G₁\odot_uv G₂ where at most one of G₁ or G₂ has 2 as Laplacian eigenvalue.

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