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The max-plus semiring Rmax is the set R∪{−∞}, equipped with the addition (a, b) 7→ max(a, b) and the multiplication (a, b) 7→ a + b. The identity element for the addition, zero, is −∞, and the identity element for the multiplication, unit, is 0. To illuminate the linear algebraic nature of the results, the generic notations +, , × (or concatenation), 0 and(More)
The study of a mixed graph and its Laplacian matrix have gained quite a bit of interest among the researchers. Mixed graphs are very important for the study of graph theory as they provide a setup where one can have directed and undirected edges in the graph. In this article we present a more general structure, namely the weighted directed graphs and supply(More)
We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we(More)
In a recent paper, Gale has given an interesting generalization of the KKM lemma in combinatorial topology. We present a similar generalization of Sperner's well-known lemma and give a constructive proof, The argument uses the familiar idea of following simplicial paths in a triangulation. To demonstrate that the algorithm must work, orientation(More)
In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old(More)
Abstract: We consider a square matrix Aǫ whose entries have first order asymptotics of the form (Aǫ)i j ∼ ai j ǫAi j when ǫ goes to 0, for some ai j ∈ C and Ai j ∈ R. We show that under a non-degeneracy condition, the order of magnitudes of the different eigenvalues ofAǫ are given by min-plus eigenvalues of min-plus Schur complements built from A = (Ai j ),(More)
Let G be a connected graph and let L(G) be its Laplacian matrix. We show that given a graph G with a point of articulation u, and a spanning tree T , there is a way to give weights to the edges of G, so that u is the characteristic vertex and the monotonicity property holds on T . A restricted graph is a graph with a restriction that each block can have at(More)