Label incidence energy of partial edge labeled graph

Let G = (V,E) be a simple graph with vertex set V = {ν₁, ν₂, ⋯, ν_n} and edge set E = {e₁, e₂, ⋯, e_m}. The label incidence matrix B_ι(G) of G is the n × m matrix whose (i, j)-entry is a if 0 labeled edge incident to 0 labeled vertex, 6 if 1 labeled edge incident to 1 labeled vertex, c if unlabeled edge incident to 0 or 1 labeled vertex and 0 otherwise. The label incidence energy IE_ι(G) is the sum of the singular values of B_ι(G). In this paper we give lower and upper bounds for IE_ι(G) in terms of graph parameters and we study label incidence energy of some families of graph.