Randić energy of digraphs

Abstract

We assume that D is a directed graph with vertex set $V\left(D\right)=\{v_1,\dots v_n\}$ and arc set $E\left(D\right)$ . A VDB topological index φ of D is defined as $\phi \left(D\right)=\frac{1}{2}\sum _{uv\in E\left(D\right)}{\phi }_{d_u^{+},d_v^{-}},$ where $d_u^{+}$ and $d_v^{-}$ denote the outdegree and indegree of vertices u and v, respectively, and ${\phi }_{i,j}$ is a bivariate symmetric function defined on nonnegative real numbers. Let $A_{\phi }=A_{\phi }\left(D\right)$ be the $n\times n$ general adjacency matrix defined as ${\left[A_{\phi }\right]}_{ij}={\phi }_{d_{v_i}^{+},d_{v_j}^{-}}$ if $v_i{v}_j\in E\left(D\right)$ , and 0 otherwise. The energy of D with respect to a VDB index φ is defined as $E_{\phi }\left(D\right)={\sum }_{i=1}^n{\sigma }_i\left(A_{\phi }\right)$ , where ${\sigma }_1\left(A_{\phi }\right)\ge {\sigma }_2\left(A_{\phi }\right)\ge \cdots \ge {\sigma }_n\left(A_{\phi }\right)\ge 0$ are the singular values of the matrix $A_{\phi }$ . We will show that in case $\phi =R$ is the Randić 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 Randić matrix has, we derive new upper and lower bounds for the Randić energy $E_R$ 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 $E_R\left(G\right)\le rank\left(G\right)$ for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.

Keywords: Digraphs; Randić energy; Randić index; VDB topological index.