Abstract. In this paper we introduce the edge-residual number ρ(G) of a graph G. We give tight upper bounds for ρ(G) in terms of the eigenvalues of the Laplacian matrix of the line graph of G. In addition, we investigate the relation between this novel parameter and the line completion number for dense graphs. We also compute the line completion number of complete bipartite graphs K_(n,m) when either m = n or both m and n are even numbers. this partially solves an open problem of Bagga, Beinecke and Varma [2].
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