Raziskovalni matematični seminar - Arhiv

 

The burning game is a two-player game on a graph, motivated by the burning and cooling processes. In this talk we will introduce the game and establish some of its basic properties, consider the Continuation Principle and its corollaries, give the general upper bound for the game burning number and comment on its relation to the burning number conjecture, and mention several other known results about the game.

Joint work with Nina Chiarelli, Marko Jakovac, William B. Kinnersley and Mirjana Mikalački.

 

Let X denote a nonempty finite set. A nonnegative matrix B\in Mat_X(R) is called  λ-doubly stochastic if

∑_{z\in X}(B)_{yz} = ∑_{z\inX}(B)_{zy}=λ for each y\in X.

Let B\in Mat_X(R) denote a normal irreducible nonnegative matrix, and B={p(B) | p\in C[t]} denote the vector space over C of all polynomials in B. For the moment let us define a 01-matrix A in the following way: (A)_{xy}=1 if and only if (B)_{xy}>0 (x,y\in X). Let Γ=Γ(A) denote a (di)graph with adjacency matrix A, diameter D, and let A_D denote the distance-D matrix of Γ. In this talk we show that B is the Bose--Mesner algebra of a commutative D-class association scheme if and only if B is a normal λ-doubly stochastic matrix with D+1 distinct eigenvalues and A_D is a polynomial in B.

This is a work in progress, and the preprint is available at  https://arxiv.org/abs/2403.00652

It is a joint work with Giusy Monzillo.