Normalized Laplacian Eigenvalues and Energy of Trees

• Published in: Taiwanese journal of mathematics Vol. 20; no. 3; pp. 491 - 507
• Main Authors: Das, Kinkar Ch, Sun, Shaowei
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.06.2016
• Subjects: Eigenvalues; Laplacians; Vertices
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.20.2016.6668

LetGbe a graph with vertex setV(G) = {v 1,v 2, …,vn } and edge setE(G). For any vertexvi ∈V(G), letdi denote the degree ofvi . The normalized Laplacian matrix of the graphGis the matrix ℒ = (ℒ ij ) given by ℒ i j = { 1 − 1 d i d j 0 if i = j and d i ≠ 0 if v i v j ∈ E ( G ) otherwise . In this paper, we obtain some bounds on the second smallest normalized Laplacian eigenvalue of treeTin terms of graph parameters and characterize the extremal trees. Utilizing these results we present some lower bounds on the normalized Laplacian energy (or Randić energy) of treeTand characterize trees for which the bound is attained. 2010Mathematics Subject Classification. 05C50. Key words and phrases. Tree, Normalized Laplacian matrix, Normalized Laplacian eigenvalues, Normalized Laplacian energy.