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Keywords

graph, Laplacian matrix, edge connectivity, integral quadratic form

Abstract

Let G be a simple connected graph with associated positive semidefinite integral quadratic form Q(x) = \sum (x(i) − x(j))^2, where the sum is taken over all edges ij of G. It is showed that the minimum positive value of Q(x) for x ∈ Z_n equals the edge connectivity of G. By restricting Q(x) to x ∈ Z_{n−1} × {0}, the quadratic form becomes positive definite. It is also showed that the number of minimal disconnecting sets of edges of G equals twice the number of vectors x ∈ Z_{n−1} ×{0} for which the form Q attains its minimum positive value.

abs_vol30_437-442.pdf (28 kB)
Abstract

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