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Yilun Shang

Publications and source records attributed to Yilun Shang.

At least 19 recordsLinked to original sources
arXiv2024

Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vuki\v{c}evi\'c Conjecture

In this work, we compute the first and second Zagreb indices for the commuting conjugacy class graphs associated with finite groups. We identify multiple classes of finite groups whose commuting conjugacy class graphs are shown to satisfy the Hansen-Vuki{\v{c}}evi{\'c} conjecture. Specifically, we prove that the conjecture holds for the commuting conjugacy class graphs of dihedral groups ($D_{2m}$), dicyclic groups, semidihedral groups, and various other two-generator groups. Moreover, we examine the case where the quotient $G/Z(G)$ is isomorphic to $D_{2m}$, $\mathbb{Z}_p \times \mathbb{Z}_p$, a Frobenius group of order $pq$ or $p^2q$, or any group of order $p^3$, for primes $p$ and $q$. In each of these cases, we demonstrate that the corresponding commuting conjugacy class graph satisfies the Hansen-Vuki{\v{c}}evi{\'c} conjecture.

math.GR↗
arXiv2024

On the spectral redundancy of pineapple graphs

In this article, we explore the concept of spectral redundancy within the class of pineapple graphs, denoted as $\mathcal{P}(α,β)$. These graphs are constructed by attaching $β$ pendent edges to a single vertex of a complete graph $K_α$. A connected graph $G$ earns the title of being spectrally non-redundant if the spectral radii of its connected induced subgraphs remain distinct. Spectral redundancy, on the other hand, arises when there is a repetition of spectral radii among the connected induced subgraphs within $G$. Specifically, we analyze the adjacency spectrum of $\mathcal{P}(α,β)$, revealing distinct eigenvalues including $0$, $-1$, and additional eigenvalues, some negative and others positive. Our investigation focuses on determining the spectral redundancy within this class of graphs, shedding light on their unique structural properties and implications for graph theory.

math.CO↗
arXiv2021

On r-noncommuting graph of finite rings

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $Γ_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y] \neq r$ and $-r$. In this paper, we study several properties of $Γ_R^r$. We show that $Γ_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of $Γ_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$.

math.RA↗
arXiv2021

Nordhaus-Guddum type results for the Steiner Gutman index of graphs

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index} $\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$ $=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $\operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $\operatorname{SGut}_k(G)+\operatorname{SGut}_k(\overline{G})$ and $\operatorname{SGut}_k(G)\cdot \operatorname{SGut}_k(\overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Δ$, minimum degree $δ$.

math.CO↗
arXiv2018

Opinion formation in multiplex networks with general initial distributions

We study opinion dynamics over multiplex networks where agents interact with bounded confidence. Namely, two neighbouring individuals exchange opinions and compromise if their opinions do not differ by more than a given threshold. In literature, agents are generally assumed to have a homogeneous confidence bound. Here, we study analytically and numerically opinion evolution over structured networks characterised by multiple layers with respective confidence thresholds and general initial opinion distributions. Through rigorous probability analysis, we show analytically the critical thresholds at which a phase transition takes place in the long-term consensus behaviour, over multiplex networks with some regularity conditions. Our results reveal the quantitative relation between the critical threshold and initial distribution. Further, our numerical simulations illustrate the consensus behaviour of the agents in network topologies including lattices and, small-world and scale-free networks, as well as for structure-dependent convergence parameters accommodating node heterogeneity. We find that the critical thresholds for consensus tend to agree with the predicted upper bounds in Theorems 4 and 5 in this paper. Finally, our results indicate that multiplexity hinders consensus formation when the initial opinion configuration is within a bounded range and, provide insight into information diffusion and social dynamics in multiplex systems modeled by networks.

physics.soc-ph↗
arXiv2017

Swarming collapse under limited information flow between individuals

The emergence of collective decision in swarms and their coordinated response to complex environments underscore the central role played by social transmission of information. Here, the different possible origins of information flow bottlenecks are identified. Using a combination of network-, control- and information-theoretic elements applied to a group of interacting self-propelled particles, the effect of varying information capacity of the signaling channel on dynamic collective behaviors is revealed. We find a sufficient condition on the information data rate that guarantees the effectiveness of swarming while also highlighting the profound connection with the topology of the underlying interaction network. We also show that when decreasing the data rate, the swarming behavior invariably vanishes following a second-order phase transition irrespective of the intrinsic noise level. The variations along the transition line are found to be in good agreement with information-theoretic predictions.

nlin.AO↗
arXiv2017

Consensus reaching in swarms ruled by a hybrid metric-topological distance

Recent empirical observations of three-dimensional bird flocks and human crowds have challenged the long-prevailing assumption that a metric interaction distance rules swarming behaviors. In some cases, individual agents are found to be engaged in local information exchanges with a fixed number of neighbors, i.e. a topological interaction. However, complex system dynamics based on pure metric or pure topological distances both face physical inconsistencies in low and high density situations. Here, we propose a hybrid metric-topological interaction distance overcoming these issues and enabling a real-life implementation in artificial robotic swarms. We use network- and graph-theoretic approaches combined with a dynamical model of locally interacting self-propelled particles to study the consensus reaching pro- cess for a swarm ruled by this hybrid interaction distance. Specifically, we establish exactly the probability of reaching consensus in the absence of noise. In addition, simulations of swarms of self-propelled particles are carried out to assess the influence of the hybrid distance and noise.

physics.bio-ph↗
arXiv2017

On the skew-spectral distribution of randomly oriented graphs

The randomly oriented graph $G_{n,p}^σ$ is an Erdős-Rényi random graph $G_{n,p}$ with a random orientation $σ$, which assigns to each edge a direction so that $G_{n,p}^σ$ becomes a directed graph. Denote by $S_n$ the skew-adjacency matrix of $G_{n,p}^σ$. Under some mild assumptions, it is proved in this paper that, the spectral distribution of $S_n$ (under some normalization) converges to the standard semicircular law almost surely as $n\rightarrow\infty$. It is worth mentioning that our result does not require finite moments of the entries of the underlying random matrix.

math.CO↗
arXiv2015

Bounds of distance Estrada index of graphs

Let $λ_1,λ_2,\cdots,λ_n$ be the eigenvalues of the distance matrix of a connected graph $G$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{λ_i}$. In this note, we present new lower and upper bounds for $DEE(G)$. In addition, a Nordhaus-Gaddum type inequality for $DEE(G)$ is given.

math.CO↗
arXiv2015

A Combinatorial Necessary and Sufficient Condition for Cluster Consensus

In this technical note, cluster consensus of discrete-time linear multi-agent systems is investigated. A set of stochastic matrices $\mathcal{P}$ is said to be a cluster consensus set if the system achieves cluster consensus for any initial state and any sequence of matrices taken from $\mathcal{P}$. By introducing a cluster ergodicity coefficient, we present an equivalence relation between a range of characterization of cluster consensus set under some mild conditions including the widely adopted inter-cluster common influence. We obtain a combinatorial necessary and sufficient condition for a compact set $\mathcal{P}$ to be a cluster consensus set. This combinatorial condition is an extension of the avoiding set condition for global consensus, and can be easily checked by an elementary routine. As a byproduct, our result unveils that the cluster-spanning trees condition is not only sufficient but necessary in some sense for cluster consensus problems.

eess.SY↗
arXiv2014

Estimating the distance Estrada index

Suppose $G$ is a simple graph on $n$ vertices. The $D$-eigenvalues $μ_1,μ_2,\cdots,μ_n$ of $G$ are the eigenvalues of its distance matrix. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{μ_i}$. In this paper, we establish new lower and upper bounds for $DEE(G)$ in terms of the Wiener index $W(G)$. We also compute the distance Estrada index for some concrete graphs including the buckminsterfullerene $C_{60}$.

math.CO↗
arXiv2014

More on the normalized Laplacian Estrada index

Let $G$ be a simple graph of order $N$. The normalized Laplacian Estrada index of $G$ is defined as $NEE(G)=\sum_{i=1}^Ne^{λ_i}$, where $λ_1,λ_2,\cdots,λ_N$ are the normalized Laplacian eigenvalues of $G$. In this paper, we give a tight lower bound for $NEE$ of general graphs. We also calculate $NEE$ for a class of treelike fractals, which contain some classical chemical trees as special cases. It is shown that $NEE$ scales linearly with the order of the fractal, in line with a best possible lower bound for connected bipartite graphs.

math.CO↗
arXiv2013

Cycles in Random Bipartite Graphs

In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle of length $t$ for all even $t\in[4,(1+o(1))n/30]$. Our theorem complements a previous result on bipancyclicity, and is closely related to a recent work of Lee and Samotij.

math.CO↗
arXiv2012

Bipancyclic subgraphs in random bipartite graphs

A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph $G(n,n,p)$ with $p(n)\gg n^{-2/3}$ asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph $G'$ of $G(n,n,p)$ with more than $(1/2+o(1))n^2p$ edges is bipancyclic. This result is tight in two ways. First, the range of $p$ is essentially best possible. Second, the proportion 1/2 of edges cannot be reduced. Our result extends a classical theorem of Mitchem and Schmeichel.

math.CO↗
arXiv2011

Focusing of Maximum Vertex Degrees in Random Faulty Scaled Sector Graphs

In this paper we study the behavior of maximum out/in-degree of binomial/Poisson random scaled sector graphs in the presence of random vertex and edge faults. We prove that the probability distribution of maximum degrees for random faulty scaled sector graphs with $n$ vertices, where each vertex spans a sector of $α$ radians, with radius $r_n\ll \sqrt{\ln n/n}$, become concentrated on two consecutive integers, under some natural assumptions of faulty probabilities.

math.CO↗
arXiv2009

Topological Properties of an Exponential Random Geometric Graph Process

In this paper, we consider a one-dimensional random geometric graph process with the inter-nodal gaps evolving according to an exponential AR(1) process, which may serve as a mobile wireless network model. The transition probability matrix and stationary distribution are derived for the Markov chains in terms of network connectivity and the number of components. We characterize an algorithm for the hitting time regarding disconnectivity. In addition, we also study topological properties for static snapshots. We obtain the degree distributions as well as asymptotic precise bounds and strong law of large numbers for connectivity threshold distance and the largest nearest neighbor distance amongst others. Both closed form results and limit theorems are provided.

cs.IT↗
arXiv2009

Degree Distributions in General Random Intersection Graphs

We study a variant of the standard random intersection graph model ($G(n,m,F,H)$) in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights, the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type.

math.CO↗