# R. B. Bapat

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)
• 2011
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)
• Discrete Mathematics
• 2008
We consider weighted graphs, where the edgeweights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained. © 2007 Elsevier B.V. All rights reserved.
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)
• 2
• 2005
R. B. Bapat 2 A. K. Lal3 Sukanta Pati 4 Abstract We consider a q-analogue of the distance matrix (called the q-distance matrix) of an unweighted tree and give formulae for the inverse and the determinant, which generalize the existing formulae for the distance matrix. We obtain the Smith normal form of the q-distance matrix of a tree. The relationship of(More)
• Graphs and Combinatorics
• 2008
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)