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 tothe randi'c matrix, here we introduce the randi'c incidence matrixof 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, therandi'c incidence energy $i_re$ of $g$ is the sum of the singularvalues of $i_r(g)$. we establish lower and upper bounds for therandi'c incidence energy. graphs for which these bounds are bestpossible are characterized. moreover, we investigate the relationbetween the randi'c incidence energy of a graph and that of itssubgraphs. also we give a sharp upper bound for the randi'cincidence energy of a bipartite graph and determine the trees withthe maximum randi'c incidence energy among all $n$-vertex trees. asa result, some results are very different from those for incidenceenergy.