Randić energy of digraphs

Abstract

We assume that D is a directed graph with vertex set V(D)={v1,…vn} and arc set E(D). A VDB topological index φ of D is defined asφ(D)=12∑uv∈E(D)φdu+,dv−, where du+ and dv− denote the outdegree and indegree of vertices u and v, respectively, and φi,j is a bivariate symmetric function defined on nonnegative real numbers. Let Aφ=Aφ(D) be the n×n general adjacency matrix defined as [Aφ]ij=φdvi+,dvj− if vivj∈E(D), and 0 otherwise. The energy of D with respect to a VDB index φ is defined as Eφ(D)=∑i=1nσi(Aφ), where σ1(Aφ)≥σ2(Aφ)≥⋯≥σn(Aφ)≥0 are the singular values of the matrix Aφ.We will show that in case φ=R is the Randić index, the spectral norm of AR is equal to 1, and rank of AR 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(G)≤rank(G) for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.

Keywords

• Randić energy
• Randić index
• VDB topological index
• Digraphs