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Maximum Spread of Vertex Degrees in a Simple Graph
Sergey Dmitrievich Onishchenko
https://arxiv.org/abs/2509.21429 https://arxiv.org/pdf/2509.21429

Maximum Spread of Vertex Degrees in a Simple Graph
We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose degrees differ by less than $k$. We aim to determine the smallest possible value $f(n,k)$ of the quantity $h_k(G)$. Interest in this question is motivated by the fact that the bipartite analogue of the problem enabled S. Cichomski and F. Petrov to prove the Burdzy…

Solving time-fractional diffusion equations with Robin boundary conditions via fractional Hamiltonian boundary value methods
Qian Luo, Aiguo Xiao, Xiaoqiang Yan, Jingmin Xia
https://arxiv.org/abs/2509.17793

Solving time-fractional diffusion equations with Robin boundary conditions via fractional Hamiltonian boundary value methods
In this paper, we propose a novel numerical scheme for solving time-fractional reaction-diffusion problems with Robin boundary conditions, where the time derivative is in the Caputo sense of order $α\in(0,1)$. The existence and uniqueness of the solution is proved. Our proposed method is based on the spectral collocation method in space and Fractional Hamiltonian boundary value methods in time. For the considered spectral collocation method, the basis functions used are not the standard polyno…

The Power Contamination Problem on Grids Revisited: Optimality, Combinatorics, and Links to Integer Sequences
El-Mehdi Mehiri, Mohammed L. Nadji
https://arxiv.org/abs/2509.12756

The Power Contamination Problem on Grids Revisited: Optimality, Combinatorics, and Links to Integer Sequences
This paper presents a combinatorial study of the power contamination problem, a dynamic variant of power domination modeled on grid graphs. We resolve a conjecture posed by Ainouche and Bouroubi (2021) by proving it is false and instead establish the exact value of the power contamination number on grid graphs. Furthermore, we derive recurrence relations for this number and initiate the enumeration of optimal contamination sets. We prove that the number of optimal solutions for specific grid fa…