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We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general ner than the spectral decomposition. As a consequence, the spectra are(More)
Based on case study of the Incheon International Airport Expressway that is the first private capital road in South Korea and the Incheon Bridge that will be open for traffic in 2010, we knew the maximization of social welfare considered toll and tax on a point of view social surplus. And considered the strategy situation that Incheon International Airport(More)
Given a graph G, let f k be the number of forests of cardinality k in G. Then the sequence (f k) has been conjectured to be unimodal for any graph G. In this paper we confirm this conjecture for Kn and Kn,n by showing that the sequence for Kn is strictly increasing (when n ≥ 4) and the sequence for Kn,n is strictly increasing except for the very last term.(More)
Let M be a finite matroid with rank function r. We will write AM when we mean that A is a subset of the ground set of M, and write M| A and MÂA for the matroids obtained by restricting M to A and contracting M on A respectively. Let M* denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial T M (x, y)(More)
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