Spectral properties of geometric–arithmetic index

• Published in: Applied mathematics and computation Vol. 277; pp. 142 - 153
• Main Authors: Rodríguez, José M., Sigarreta, José M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 20.03.2016
• Subjects: Algebra; Computation; Geometric–arithmetic index; Graph invariant; Graph theory; Graphs; Laplacian eigenvalues; Laplacian matrix; Mathematical analysis; Mathematical models; Matrices (mathematics); Spectra; Spectral properties; Topological index; 92E10; Geometric–arithmetic index; Topological index; 05C50; Spectral properties; 05C07; Laplacian matrix; Graph invariant; Laplacian eigenvalues; 15A18
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2015.12.046

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the geometric–arithmetic index GA1 from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices. Moreover, using this matrix, we define a GA Laplacian matrix which determines the geometric–arithmetic index of a graph and satisfies properties similar to the ones of the classical Laplacian matrix.