Randi c 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 Randi c matrix, here we introduce the Randi c 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 Randi c 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 Randi c incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randi c incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randi c incidence energy of a bipartite graph and determine the trees with the maximum Randi c incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.