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Exact Expectation and Variance of Minimal Basis of Random Matroids

Wojciech Kordecki, Anna Lyczkowska-Hanćkowiak (2013)

Discussiones Mathematicae Graph Theory

We formulate and prove a formula to compute the expected value of the minimal random basis of an arbitrary finite matroid whose elements are assigned weights which are independent and uniformly distributed on the interval [0, 1]. This method yields an exact formula in terms of the Tutte polynomial. We give a simple formula to find the minimal random basis of the projective geometry PG(r − 1, q).

Extremal Matching Energy of Complements of Trees

Tingzeng Wu, Weigen Yan, Heping Zhang (2016)

Discussiones Mathematicae Graph Theory

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...

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