A sharp upper bound on the incidence energy of graphs in terms of connectivity

Let G=(V(G),E(G)) be a simple undirected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G)={e1,e2,…,em}. The incidence matrix I(G) of G is the n×m matrix whose (i,j)-entry is 1 if vi is incident to ej and 0 otherwise. The incidence energy IE(G) of G is the sum of the singular values of I(G). Let Ks be the complete graph on s vertices. In this paper, we derive an upper bound for the incidence energy of the graphs G on n vertices having a vertex connectivity less than or equal to k. This upper bound is attained if and only if G=Kk∨(K1∪Kn-k-1) obtained from the graphs Kk and K1∪Kn-k-1 and the edges connecting each vertex of Kk with every vertex of K1∪Kn-k-1.