RESUMO
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.
RESUMO
In this article, we study the degree-based topological indices in a random polyomino chain. The key purpose of this manuscript is to obtain the asymptotic distribution, expected value and variance for the degree-based topological indices in a random polyomino chain by using a martingale approach. Consequently, we compute the degree-based topological indices in a polyomino chain, hence some known results from the existing literature about polyomino chains are obtained as corollaries. Also, in order to apply the results, we obtain the expected value of several degree-based topological indices such as Sombor, Forgotten, Zagreb, atom-bond-connectivity, Randic and geometric-arithmetic index of a random polyomino chain.