Randić energy of digraphs

We assume that D is a directed graph with vertex set ⁡()={1,…} and arc set ⁡(). A VDB topological index φ of D is defined as ⁡()= 1 2 ∑ ⁢∈⁡() + ,−, where + and − denote the outdegree and indegree of vertices u and v, respectively, and , is a bivariate symmetric function defined on nonnegative real numbers. Let =⁡() be the × general adjacency matrix defined as []⁢=+,− if ⁢∈⁡(), and 0 otherwise. The energy of D with respect to a VDB index φ is defined as ℰ⁡()=∑ =1⁡(), where 1⁡()≥2⁡()≥⋯≥⁡()≥0 are the singular values of the matrix .We will show that in case =ℛ is the Randić index, the spectral norm of ℛ is equal to 1, and rank of ℛ is equal to rank of the adjacency matrix of D. Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randić matrix has, we derive new upper and lower bounds for the Randić energy ℰℛ in digraphs. Some of these generalize known results for the Randić energy of graphs. Also, we deduce a new upper bound for the Randić energy of graphs in terms of rank, concretely, we show that ℰℛ⁡()≤⁢⁢⁢⁡() for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.

• Autores: Rada Rincón, Juan Pablo; Cruz Rodes, Roberto; Monsalve Aristizabal, Juan Daniel
• Tipo de recurso: Article of investigation
• Fecha de publicación: 2022
• Institución: Universidad de Antioquia
• Repositorio: Repositorio UdeA
• Idioma: eng
• OAI Identifier: oai:bibliotecadigital.udea.edu.co:10495/46335
• Acceso en línea: https://hdl.handle.net/10495/46335
• Palabra clave: Teoría de grafos; Graph theory; Teoría de conjuntos; Set theory; Matrices (Matemáticas); Matrices; Energía de Randic; Randić energy
• Rights: openAccess
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/