arXiv subjects

Wensheng Sun

Publications and source records attributed to Wensheng Sun.

10 recordsLinked to original sources

arXiv2026

Combinatorial explanation of the weighted Kirchhoff index of graphs

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $\omega:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $\omega(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to $2$-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.

math.CO↗

arXiv2026

Resistance Curvature: Recognition, Polyhedral Structure, and Graph Products

Resistance curvature, introduced by Devriendt and Lambiotte, is a novel discrete curvature notion defined through effective resistance. A graph is called resistance nonnegative if there exists a choice of positive edge weights for which the resistance curvature is nonnegative at every vertex. This property has a notable combinatorial interpretation in terms of random spanning trees: a graph is resistance nonnegative if and only if it admits a distribution on its spanning trees under which every vertex has expected degree at most two. The resistance nonnegativity can also be characterized by the tree double matching polytope. These characterizations reveal strong connections among resistance curvature, effective resistance, spanning tree distributions, matching theory, and polyhedral combinatorics. Based on the sign of the curvature, Devriendt introduced the classes of resistance nonnegative (RN), resistance positive (RP), and strictly resistance nonnegative (SRN) graphs, and posed several questions concerning their recognition, polyhedral structure, and structural properties. In this paper, we first answer Devriendt's question on the computational complexity of recognizing RN, RP, and SRN graphs by proving that all three classes can be recognized in polynomial time. We then address his question concerning the tree double matching polytope $\Theta(G)$. Further, we characterize the vertices of $\Theta(G)$ in terms of full-rank systems of tight constraints. Whenever $\Theta(G)\neq\emptyset$, we also determine the least positive integer $k_G$ such that $k_G\Theta(G)$ is a lattice polytope. Finally, for every finite Cartesian product of paths, we explicitly construct an average point satisfying the condition for resistance nonnegativity, thereby obtaining that such graphs are RN. We further characterize the classes of such Cartesian product graphs that are RP or SRN.

math.CO↗

arXiv2026

On the Lei--Bai conjecture on $5$-regular Lin--Lu--Yau Ricci-flat graphs

We study the Ricci curvature introduced by Lin, Lu, and Yau. A graph is called Ricci-flat if every edge has curvature zero. Lei and Bai classified $5$-regular symmetric Ricci-flat graphs by proving that every such graph is isomorphic to a particular $72$-vertex graph $\RF$, and conjectured that every $5$-regular Ricci-flat graph is either isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. In this paper, we disprove this conjecture by constructing an infinite family of connected $5$-regular Ricci-flat graphs, none of which is isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. This shows that the conjectured extension of the classification from the symmetric setting to general $5$-regular Ricci-flat graphs fails and that the class of such graphs is substantially richer than previously conjectured. To establish these results, we use an optimal-assignment formulation of Lin--Lu--Yau curvature to verify the Ricci-flatness of the constructed graphs.

math.CO↗

arXiv2026

On two conjectures concerning Kemeny's constant of graphs

Kemeny's constant for a connected graph $G$, denoted by $\mathcal{K}(G)$, is the expected time for a random walk to reach a randomly chosen vertex $u$, regardless of the choice of the initial vertex. Recently, Kim et al. (2026) proposed two conjectures on Kemeny's constant. The first conjecture asserts that if $G$ is a connected graph of order $n$ and diameter 2, then $\mathcal{K}(G) = O(n)$. The second conjecture asserts that if $G$ be a graph of order $n$, then $\min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} = O(n)$, and if both $G$ and $\overline{G}$ are connected, then $\mathcal{K}(G)\mathcal{K}(\overline{G}) = O(n^4)$, where $\overline{G}$ denotes the complement of $G$. In this paper, we confirm both conjectures. For the first conjecture, we prove that if $G$ is a connected graph of order $n$ and diameter 2, then \[ \mathcal{K}(G) \leq (3 + \sqrt{5})(n - 1). \] For the second conjecture, we prove that for any $n$-vertex graph $G$, \[ \min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} \leq (8 + 2\sqrt{5})n - (10 + 2\sqrt{5}). \] Moreover, if both $G$ and $\overline{G}$ are connected, then \[ \mathcal{K}(G)\mathcal{K}(\overline{G}) \leq \frac{3 + \sqrt{5}}{2}n^4. \] Our proof relies on effective estimates on resistance distances and spectral gaps of graphs.

math.CO↗

arXiv2025

A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes

A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching.

math.CO↗

arXiv2025

On the minimum constant resistance curvature conjecture of graphs

Let $G$ be a connected graph with $n$ vertices. The resistance distance $Ω_{G}(i,j)$ between any two vertices $i$ and $j$ of $G$ is defined as the effective resistance between them in the electrical network constructed from $G$ by replacing each edge with a unit resistor. The resistance matrix of $G$, denoted by $R_G$, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $Ω_{G}(i,j)$. The resistance curvature $κ_i$ in the vertex $i$ is defined as the $i$-th component of the vector $(R_G)^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes the all-one vector. If all the curvatures in the vertices of $G$ are equal, then we say that $G$ has constant resistance curvature. Recently, Devriendt, Ottolini and Steinerberger \cite{kde} conjectured that the cycle $C_n$ is extremal in the sense that its curvature is minimum among graphs with constant resistance curvature. In this paper, we confirm the conjecture. As a byproduct, we also solve an open problem proposed by Xu, Liu, Yang and Das \cite{kxu} in 2016. Our proof mainly relies on the characterization of maximum value of the sum of resistance distances from a given vertex to all the other vertices in 2-connected graphs.

math.CO↗

arXiv2024

Solution to a conjecture on resistance distances of block tower graphs

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. The resistance diameter of $G$, denoted by $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture.

math.CO↗

arXiv2022

Multi-objective Optimization of Notifications Using Offline Reinforcement Learning

Mobile notification systems play a major role in a variety of applications to communicate, send alerts and reminders to the users to inform them about news, events or messages. In this paper, we formulate the near-real-time notification decision problem as a Markov Decision Process where we optimize for multiple objectives in the rewards. We propose an end-to-end offline reinforcement learning framework to optimize sequential notification decisions. We address the challenge of offline learning using a Double Deep Q-network method based on Conservative Q-learning that mitigates the distributional shift problem and Q-value overestimation. We illustrate our fully-deployed system and demonstrate the performance and benefits of the proposed approach through both offline and online experiments.

cs.LG↗

arXiv2019

Impulsive Noise Mitigation in Underwater Acoustic Communication Systems: Experimental Studies

Impulsive noise is a major impediment to orthogonal frequency-division multiplexing (OFDM) based underwater acoustic (UWA) communications. In this work, we evaluate the performance of a memoryless analog nonlinear preprocessor (MANP) that is used to mitigate outliers. The proposed MANP exhibits intermittent nonlinearity only in the presence of the impulsive noise and suppresses the power of outliers based on their amplitudes. Since the outliers are distinguishable in the analog domain prior to anti-aliasing filtering, the MANP outperforms its digital counterparts in all scenarios. Experimental results using data collected in an under-ice environment, demonstrate the superior BER performance of our approach relative to classical nonlinear approaches such as blanking and clipping.

eess.SP↗

arXiv2017

Multi-speaker Recognition in Cocktail Party Problem

This paper proposes an original statistical decision theory to accomplish a multi-speaker recognition task in cocktail party problem. This theory relies on an assumption that the varied frequencies of speakers obey Gaussian distribution and the relationship of their voiceprints can be represented by Euclidean distance vectors. This paper uses Mel-Frequency Cepstral Coefficients to extract the feature of a voice in judging whether a speaker is included in a multi-speaker environment and distinguish who the speaker should be. Finally, a thirteen-dimension constellation drawing is established by mapping from Manhattan distances of speakers in order to take a thorough consideration about gross influential factors.

eess.AS↗