Narrow Results

A new notion of vertex independence and rank for a finite graph G is introduced. The independence of vertices is based on the boolean independence of columns of a natural boolean matrix associated to G. Rank is the cardinality of the largest set of independent columns. Some basic properties and some more advanced theorems are proved. Geometric properties of the graph are related to its rank and independent sets.

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation $\mathrm{\Delta }\psi +f(\psi )=0$ with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.

This paper is devoted to the study of the behaviors of the solution to Fujita-type equations on finite graphs. Under certain conditions given by absorption term of the equations, we prove respectively local existence and blow-up results of solutions to Fujita-type equations on finite graphs. Our results contain some previous results as special cases. Finally, we provide some numerical experiments to illustrate the applicability of the obtained results.

In this paper, we study the blow-up problem for Fujita-type equations with a general absorption term $f$ on finite graphs and locally finite graphs, respectively. We prove that if $f$ satisfies appropriate conditions, then the solutions of the equation blow up in finite time. The obtained results are generalization of those given by the author in previous papers. At the end of the paper, we provide some numerical experiments to illustrate the applicability of the obtained results.