Randiʹʹc incidence energy of graphs

Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots , v_n}$ and edge set $E(G) = {e_1, e_2,ldots , e_m}$. Similar to the Randiʹc matrix, here we introduce the Randiʹc incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $ntimes m$ matrix whose $(i, j)$-entry is $(d_i)^{-frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randiʹc incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randiʹc incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randiʹc incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randiʹc incidence energy of a bipartite graph and determine the trees with the maximum Randiʹc incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.