Oriented unicyclic graphs with minimal skew Randić energy

Abstract

Let $G$ be a simple graph with vertex set $V(G)=\{v_{1},v_{2},$ $\dots,v_{n}\}$, and $G^{\sigma}$ be an orientation of $G$. Denote by $d(v_i)$ the degree of the vertex $v_i$ for $i=1,2,\dots,n$. The skew Randić matrix of $G^{\sigma}$, denoted by $R_S(G^{\sigma})$, is the real skew-symmetric matrix $(r_{ij})_{n\times n}$, where $r_{ij}={1}/{\sqrt{d(v_i)d(v_j)}}$ and $r_{ji}=-{1}/{\sqrt{d(v_i)d(v_j)}}$ if $v_i\rightarrow v_j$ is an arc of $G^{\sigma}$, otherwise $r_{ij}=r_{ji}=0$. The skew Randi\'{c} energy $\mathcal{RE}_S(G^{\sigma})$ of $G^{\sigma}$ is defined as the sum of the norms of all the eigenvalues of $R_S(G^{\sigma})$. In this paper, the oriented unicyclic graphs with minimal skew Randić energy are determined.