Extremal graphs for the sum of the two largest signless Laplacian\n eigenvalues

2013/10/31 by Carla Silva Oliveira, Leonardo de Lima, Oliveira, Carla Silva +5
Chemistry · Materials Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Nanocluster Synthesis and Applications #Spectral Theory (math.SP) #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.1310.8559

openalex publication_date 2013/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple graph on n vertices and e(G) edges. Consider Q(G) = D\n+ A as the signless Laplacian of G, where A is the adjacency matrix and\nD is the diagonal matrix of the vertices degree of G. Let q₁(G) and\nq₂(G) be the first and the second largest eigenvalues of Q(G),\nrespectively, and denote by S_n⁺ the star graph plus one edge. In this\npaper, we prove that inequality q₁(G)+ q₂(G) <= e(G)+3 is tighter for the\ngraph S_n⁺ among all firefly graphs and also tighter to S_n⁺ than\nto the graphs K_k vee \K_n-k recently presented by Ashraf,\nOmidi and Tayfeh-Rezaie. Also, we conjecture that the same inequality is\ntighter to S_n⁺ than any other graph on n vertices.\n

Related

• Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues
• On the sum of the first two largest signless Laplacian eigenvalues of a graph
• On a conjecture for the signless Laplacian eigenvalues
• On two conjectures on sum of the powers of signless Laplacian eigenvalues of a graph
• A lower bound for the sum of the two largest signless Laplacian eigenvalues