arXiv subjects

Xianhao Meng

Publications and source records attributed to Xianhao Meng.

4 recordsLinked to original sources

arXiv2026

Resolvent comparison for Hermitian block matrices and Kelmans transformations

A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block. An orthogonal change of basis and Schur complementation give an exact formula for the compressed resolvent difference as a difference of two rank-one matrices. This formula characterizes admissible coupling subspaces by proportionality of projected resolvent vectors. Polynomial coefficient kernels describe all such subspaces, and cyclic subspaces give dimension bounds and a maximum-dimension result for a two-block family. We also prove a general block comparison theorem and give monotonicity and equality criteria for nonnegative rank-one couplings. Applications concern graph joins, complex attachments, and Hermitian adjacency matrices.

math.RA↗

arXiv2026

Small and large quasi-kernels: a weighted transfer theorem

A quasi-kernel of a digraph $D$ is an independent set from which every vertex is reachable by a directed path of length at most two. The Small Quasi-kernel Conjecture asserts that every source-free digraph on $n$ vertices has a quasi-kernel of size at most $n/2$, whereas the Large Quasi-kernel Conjecture asks for a quasi-kernel whose closed out-neighbourhood has size at least $n/2$. Spiro proved that the former implies the latter. In this paper, we establish a weighted transfer theorem: if every nonnegative integer weighting of a source-free digraph $D$ admits a quasi-kernel whose closed out-neighbourhood contains at least a $θ$-fraction of the total weight, then $D$ has a quasi-kernel of size at most $\lfloor(1-θ)α(D)+θμ\rfloor$ for $0<θ\le1/2$, where $μ$ is the maximum matching size between a maximum independent set and its complement. Using the independent blow-up construction of Spiro, we deduce that the universal small and large quasi-kernel assertions are equivalent for every such $θ$, with no loss in the parameter. In particular, the two conjectures are equivalent. As an application, we prove the half-order bound for source-free break digraphs.

math.CO↗

arXiv2023

Approximate Nash Equilibria Algorithms for Shapley Network Design Games

We consider a weighted Shapley network design game, where selfish players choose paths in a network to minimize their cost. The cost function of each edge in the network is affine linear with respect to the sum of weights of the players who choose the edge. We first show the existence of α-approximate pure Nash equilibrium by constructing a potential function and establish an upper bound O(log2(W)) of α, where W is the sum of the weight of all players. Furthermore, we assume that the coefficients of the cost function (affine linear function) of the edge all are ϕ-smooth random variables on [0, 1]. In this case, we show that ε-best response dynamics can compute the (1 + ε)α-approximate pure Nash equilibrium (εis a positive constant close to 0) in polynomial time by proving the expected number of iterations is polynomial in 1/ε, ϕ, the number of players and the number of edges in the network.

cs.GT↗

arXiv2021

On critical graphs for the chromatic edge-stability number

The {\em chromatic edge-stability number} $es_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em $(3,2)$-critical} if $χ(G)=3$, $es_χ(G)=2$ and for any edge $e\in E(G)$, $es_χ(G-e)<es_χ(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].

math.CO↗