Randic energy of digraphs.
Heliyon
; 8(11): e11874, 2022 Nov.
Article
em En
| MEDLINE
| ID: mdl-36458296
We assume that D is a directed graph with vertex set V ( D ) = { v 1 ,
v n } and arc set E ( D ) . A VDB topological index φ of D is defined as φ ( D ) = 1 2 ∑ u v ∈ E ( D ) φ d u + , d v - , where d u + and d v - denote the outdegree and indegree of vertices u and v, respectively, and φ i , j is a bivariate symmetric function defined on nonnegative real numbers. Let A φ = A φ ( D ) be the n × n general adjacency matrix defined as [ A φ ] i j = φ d v i + , d v j - if v i v j ∈ E ( D ) , and 0 otherwise. The energy of D with respect to a VDB index φ is defined as E φ ( D ) = ∑ i = 1 n σ i ( A φ ) , where σ 1 ( A φ ) ≥ σ 2 ( A φ ) ≥ ⯠≥ σ n ( A φ ) ≥ 0 are the singular values of the matrix A φ . We will show that in case φ = R is the Randic index, the spectral norm of A R is equal to 1, and rank of A 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 Randic matrix has, we derive new upper and lower bounds for the Randic energy E R in digraphs. Some of these generalize known results for the Randic energy of graphs. Also, we deduce a new upper bound for the Randic energy of graphs in terms of rank, concretely, we show that E R ( G ) ≤ r a n k ( G ) for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.
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1
Coleções:
01-internacional
Base de dados:
MEDLINE
Idioma:
En
Revista:
Heliyon
Ano de publicação:
2022
Tipo de documento:
Article
País de afiliação:
Colômbia
País de publicação:
Reino Unido