Randi？ 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 $n\times m$ 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.