Geodesic
Extremal Matching Energy of Complements of Trees
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 505-521

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Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . ., n/2. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy.
Keywords: matching polynomial, matching energy, Hosoya index, energy 
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Wu, Tingzeng; Yan, Weigen; Zhang, Heping. Extremal Matching Energy of Complements of Trees. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 505-521. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a0/