The least signless Laplacian eigenvalue of the complements of bicyclic\n graphs

2019/07/10 by Xiaoyun Feng, Feng, Xiaoyun, Guoping Wang +1
Mathematics · Chemistry · Materials Science · #Graph theory and applications #Synthesis and Properties of Aromatic Compounds #Magnetism in coordination complexes

paper · pdf · doi:10.48550/arxiv.1907.04798

Abstract

Suppose that G is a connected simple graph with the vertex set V(G)= v₁,\nv₂,\⋯,v_n . Then the adjacency matrix of G is A(G)=(a_ij)_n\×\nn, where a_ij=1 if v_i is adjacent to v_j, and otherwise a_ij=0.\nThe degree matrix D(G)=diag(d_G(v₁), d_G(v₂), \…, d_G(v_n)), where\nd_G(v_i) denotes the degree of v_i in the graph G (1\≤ i\≤ n).\nThe matrix Q(G)=D(G)+A(G) is called the signless Laplacian matrix of G. The\nleast eigenvalue of Q(G) is also called the least signless Laplacian\neigenvalue of G. In this paper we give two graft transformations and then use\nthem to characterize the unique connected graph whose least signless Laplacian\neigenvalue is minimum among the complements of all bicyclic graphs.\n

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