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

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