Laplacian Spectral Radius Research Articles

The effect on the (signless Laplacian) spectral radii of uniform hypergraphs by subdividing an edge

In this paper, we investigate how the spectral radius (resp., signless Laplacian spectral radius) changes when a connected uniform hypergraph is perturbed by subdividing an edge. We extend the results of Hoffman and Smith from connected graphs to connected uniform hypergraphs. Moreover, we also study how the Laplacian spectral radius behaves when an odd-bipartite uniform hypergraph is perturbed by subdividing an edge. As applications, we determine the unique unicyclic hypergraph with the largest signless Laplacian spectral radius, and also determine the unique unicyclic even uniform hypergraph with the largest Laplacian spectral radius.

On sufficient spectral radius conditions for hamiltonicity

During the last decade several research groups have published results on sufficient conditions for the hamiltonicity of graphs in terms of their spectral radius and their signless Laplacian spectral radius. Here we extend some of these results. All of our results involve the characterization of the exceptional graphs, i.e., all the nonhamiltonian graphs that satisfy the condition. The proofs of our main results are based on the Bondy–Chvátal closure, a degree sequence condition due to Chvátal, and an operation on the edges that is known as Kelmans’ transformation.

A sharp upper bound on the spectral radius of a nonnegative k-uniform tensor and its applications to (directed) hypergraphs

In this paper, we obtain a sharp upper bound on the spectral radius of a nonnegative k-uniform tensor and characterize when this bound is achieved. Furthermore, this result deduces the main result in [X. Duan and B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. 439:2961–2970, 2013] for nonnegative matrices; improves the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph for some known results in [D.M. Chen, Z.B. Chen and X.D. Zhang, Spectral radius of uniform hypergraphs and degree sequences, Front. Math. China 6:1279–1288, 2017]; and presents some new sharp upper bounds for the adjacency spectral radius and signless Laplacian spectral radius of a uniform directed hypergraph. Moreover, a characterization of a strongly connected k-uniform directed hypergraph is obtained.

Signless Laplacian spectral conditions for Hamilton-connected graphs with large minimum degree

In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.

The signless Laplacian spectral radius of graphs with forbidding linear forests

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k⋅P3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k⋅P3 as a subgraph, respectively.

Some spectral sufficient conditions for a graph being pancyclic

Let $G(V, E)$ be a simple connected graph of order $n$. A graph of order $n$ is called pancyclic if it contains all the cycles $C_k$ for $k\in \{3, 4, \cdot\cdot\cdot, n\}$. In this paper, some new spectral sufficient conditions for the graph to be pancyclic are established in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.

New bounds on the distance Laplacian and distance signless Laplacian spectral radii

Let G be a simple undirected connected graph. In this paper, new upper bounds on the distance Laplacian spectral radius of G are obtained. Moreover, new lower and upper bounds for the distance signless Laplacian spectral radius of G are derived. Some of the above mentioned bounds are sharp and the graphs attaining the corresponding bound are characterized. Several illustrative examples are included.

Laplacian eigenvalues of the zero divisor graph of the ring [formula omitted

We study the Laplacian eigenvalues of the zero divisor graph Γ(Zn) of the ring Zn and prove that Γ(Zpt) is Laplacian integral for every prime p and positive integer t≥2. We also prove that the Laplacian spectral radius and the algebraic connectivity of Γ(Zn) for most of the values of n are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of n. The values of n for which algebraic connectivity and vertex connectivity of Γ(Zn) coincide are also characterized.

On the normalized Laplacian spectral radius, Laplacian incidence energy and Kemeny's constant

Let G=(V,E) be a simple connected graph of order n and size m, and let ρ1≥ρ2≥⋯≥ρn−1>ρn=0, be the normalized Laplacian eigenvalues of G. The greatest eigenvalue, ρ1, is referred to as the normalized Laplacian spectral radius. The Laplacian incidence energy, the Randić energy, and Kemeny's constant of G, are defined by LIE(G)=∑i=1n−1ρi, RE(G)=∑i=1n|ρi−1|, and K(G)=∑i=1n−11ρi, respectively. New lower bounds for the normalized Laplacian spectral radius are obtained. The acquired bounds are then used to obtain new bounds for the LIE(G). In addition, some relations between LIE(G), K(G), RE(G) are determined.

Laplacian Spectra of Power Graphs of Certain Finite Groups

In this article, various aspects of Laplacian spectra of power graphs of finite cyclic, dicyclic and finite p-groups are considered. The algebraic connectivity is studied and the multiplicity of the Laplacian spectral radius is determined completely for power graphs of all of these groups. Then the equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above groups. Orders of dicyclic groups, for which their power graphs are Laplacian integral, are determined. Moreover, it is proved that the notion of equality of the vertex connectivity and the algebraic connectivity and the notion of Laplacian integral are equivalent for power graphs of dicyclic groups. All possible values of Laplacian eigenvalues are obtained for power graphs of finite p-groups. This shows that power graphs of finite p-groups are Laplacian integral.

On the signless Laplacian spectral radius of Ks,t -minor free graphs

ABSTRACT In this paper, we prove that if G is a -minor free graph of order with , the signless Laplacian spectral radius with equality if and only if and , where for n − s + 1=pt + r and . In particular, if t=3 and , then is the unique -minor free graph of order n with the maximum signless Laplacian spectral radius. In addition, is the unique extremal graph with the maximum signless Laplacian spectral radius among all -minor free graphs of order .

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G) = {v1, v2,…, vn}. The transmission Tr(vi) (or Di) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n × n diagonal matrix with its (i, i)-entry equal to TrG(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as $$\mathscr{Q}(G)=\operatorname{Tr}(G)+D(G)$$ , where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.

The (distance) signless Laplacian spectral radius of digraphs with given arc connectivity

Let G‾n,k denote the set of strongly connected digraphs with order n and arc connectivity k, and let G‾n,k⁎ denote the set of digraphs in G‾n,k with all vertices having outdegree and indegree greater than k. In this paper, we determine the unique digraph with the maximum signless Laplacian spectral radius among all digraphs in G‾n,k. We also determine the unique one with the maximum signless Laplacian spectral radius among all digraphs in G‾n,k⁎ with k=1,2. For the general case, we propose a conjecture on the maximum signless Laplacian spectral radius among all digraphs in G‾n,k⁎. Moreover, we characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k. We also characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k⁎ with k=1,2. For the general case, we propose a conjecture on the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k⁎.

Model reduction of synchronized homogeneous Lur’e networks with incrementally sector-bounded nonlinearities

This paper proposes a model order reduction scheme that reduces the complexity of diffusively coupled homogeneous Lur’e systems. We aim to reduce the dimension of each subsystem and meanwhile preserve the synchronization property of the overall network. Using the Laplacian spectral radius, we characterize the robust synchronization of the Lur’e network by a linear matrix inequality (LMI), whose solutions then are treated as generalized Gramians for the balanced truncation of the linear component of each Lur’e subsystem. It is verified that, with the same communication topology, the resulting reduced-order network system is still robustly synchronized, and an a priori bound on the approximation error is guaranteed to compare the behaviors of the full-order and reduced-order Lur’e subsystems.

Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph

‎‎For a simple connected graph $G$ with $n$ vertices and $m$ edges‎, ‎let $overrightarrow{G}$ be a digraph obtained by giving an arbitrary direction to the edges of $G$‎. ‎In this paper‎, ‎we consider the skew Laplacian/skew adjacency matrix of the digraph $overrightarrow{G}$‎. ‎We obtain upper bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of various parameters (like oriented degree‎, ‎average oriented degree) associated with the structure of the digraph $overrightarrow{G}$‎. ‎We also obtain upper and lower bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of skew Laplacian/skew adjacency rank of the digraph $overrightarrow{G}$‎.

The spectral radius and domination number in linear uniform hypergraphs

This paper investigates the spectral radius and signless Laplacian spectral radius of linear uniform hypergraphs. A dominating set in a hypergraph H is a subset D of vertices if for every vertex v not in D there exists $$u\in D$$ such that u and v are contained in a hyperedge of H. The minimum cardinality of a dominating set of H is called the domination number of H. We present lower bounds on the spectral radius and signless Laplacian spectral radius of a linear uniform hypergraph in terms of its domination number.

Spectral results on Hamiltonian problem

Let α be a non-negative real number, and let Θ(G,α) be the largest eigenvalue of A(G)+αD(G). Specially, Θ(G,0) and Θ(G,1) are called the spectral radius and signless Laplacian spectral radius of G, respectively. A graph G is said to be Hamiltonian (traceable) if it contains a Hamiltonian cycle (path), and a graph G is called Hamilton-connected if any two vertices are connected by a Hamiltonian path in G. The number of edges of G is denoted by e(G). Recently, the (signless Laplacian) spectral property of Hamiltonian (traceable, Hamilton-connected) graphs received much attention. In this paper, we shall give a general result for all these existed results. To do this, we first generalize the concept of Hamiltonian, traceable, and Hamilton-connected to s-suitable, and we secondly present a lower bound for e(G) to confirm the existence of s-suitable graphs. Thirdly, when 0≤α≤1, we obtain a lower bound for Θ(G,α) to confirm the existence of s-suitable graphs. Consequently, our results generalize and improve all these existed results in this field, including the main results of Chen et al. (2018), Feng et al. (2017), Füredi et al. (2017), Ge et al. (2016), Li et al. (2016), Nikiforov et al. (2016), Wei et al. (2019) and Yu et al. (2013, 2014).

Sharp upper bounds for the adjacency and the signless Laplacian spectral radius of graphs

Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has at least one common adjacent vertex. Our results improve some known upper bounds. The main tool we use here is the Lagrange identity.

Bounds for the skew Laplacian spectral radius of oriented graphs

We consider the skew Laplacian matrix of a digraph −→G obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph −→G and characterize the extremal graphs.

Spectral radius and Hamiltonicity of graphs

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement respectively.