Energy of weighted digraphs (Q2030430)

The purpose of the paper is to study the spectra of a weighted digraph, where weights are nonzero real numbers. So every digraph can be regarded as the weighted digraph with weight of each arc equal to 1. The weight of a weighted subdigraph D1 of a weighted graph D is defined as the product of weights of the arcs of D1. D1 is said to be positive or negative depending on whether or not its weight is positive or negative. A weighted digraph is said to be balanced if each of its cycles has positive weight, and unbalanced, otherwise. A weighted digraph is said to be linear if its components are cycles. The paper gives formulas for the characteristic polynomial of two families of weighted bipartite digraphs and studies the sign alternating property of the coefficients of the characteristic polynomial in some classes of weighted digraphs. The paper also extends the notion of energy to weighted digraphs and obtains Coulson's integral formula, thereby enabling energy comparison property by means of a quasi-order relation, and characterizing unicycle weighted digraphs having minimum and maximum energy. Finally, the paper derives McClelland upper bound for the energy weighted digraphs as well as an upper bound for the energy of a weighted digraph in terms of number of arcs and their weights.