Signless Laplacian Spectral Radius Research Articles

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1,$ respectively, which intersect in exactly one common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

(Distance) signless Laplacian spectral radius of graphs with fixed fractional matching number

(Distance) signless Laplacian spectral radius of graphs with fixed fractional matching number

The Q-minimizer graph with given independence number

Let Gn,α be the set of all connected graphs of order n with independence number α. A graph is called the Q-minimizer graph (A-minimizer graph) if it attains the minimum signless Laplacian spectral radius (adjacency spectral radius) over all graphs in Gn,α. In this paper, we first show that the Q-minimizer graph must be a tree for α≥⌈n2⌉, and then we derive seven propositions about the Q-minimizer graph. Moreover, when n−α is a constant, the structure of the Q-minimizer graph is characterized. The method of getting Q-minimizer graph in this paper is different from that of getting A-minimizer graph. As applications, we determine the Q-minimizer graphs for α=n−1,n−2,n−3 and n−4, respectively. The results of α=n−1,n−2,n−3 are consistent with that in Li and Shu (2010) [15] and the result of α=n−4 is new. Interestingly, the Q-minimizer graph in Gn,n−4 is unique, which is exactly one of the A-minimizer graphs in Gn,n−4.

Some extremal problems on [formula omitted]-spectral radius of graphs with given size

Nikiforov defined the Aα-matrix of a graph G as Aα(G)=αD(G)+(1−α)A(G), where α∈[0,1], D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of Aα(G) is called the Aα-spectral radius of G, denoted by ρα(G). In this paper, we first give an upper bound on ρα(G) of a connected graph G with fixed size m≥3k and maximum degree Δ≤m−k, where k is a positive integer. For two connected graphs G1 and G2 with size m≥4, employing this upper bound, we prove that ρα(G1)>ρα(G2) if Δ(G1)>Δ(G2) and Δ(G1)≥2m3+1. As an application, we determine the graph with the maximal Aα-spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph.

On the relationship between shortlex order and Aα-spectral radii of graphs with starlike branch tree

For graph G, let ρα(G) denote its Aα-spectral radius and let Gv(n1,n2,…,nd) be the graph obtained from G by appending d paths on n1,n2,…,nd vertices respectively at the vertex v of G. For connected graph G with at least two vertices, the famous Li-Feng Grafting Theorem indicates that ρ0(Gv(p+1,q−1))<ρ0(Gv(p,q)) holds for any p≥q≥1. Later, Cvetković and Simić showed that the similar inequality also holds for the signless Laplacian spectral radius (equivalently, the A0.5-spectral radius). Confirming a Nikiforov-Rojo Conjecture, Lin, Huang and Xue (independently, Guo and Zhou) discovered that this phenomenon happens for Aα-spectral radius if 0≤α<1. Recently, Oliveira, Stevanović and Trevisan proved that the spectral radius has similar property for graphs obtained by appending at least three paths or a varying number of paths to an isolated vertex. In this note, we extend these former results by showing that the Aα-spectral radius of Gv(n1,n2,…,nd) will increase according to the shortlex ordering of (n1,n2,…,nd).

Maximizing the signless Laplacian spectral radius of minimally 3-connected graphs with given size

A 3-connected graph G is minimally 3-connected if it becomes not 3-connected after deleting any edge of G. In this paper, we give an upper bound on the signless Laplacian spectral radius of minimally 3-connected graphs with m edges, and completely characterize the corresponding extremal graph in the case when m is even. As a corollary, we determine the unique graph with the maximal signless Laplacian spectral radius among all Halin graphs with m edges.

New Improved Bounds for Signless Laplacian Spectral Radius and Nordhaus-Gaddum Type Inequalities for Agave Class of Graphs

Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determine the bounds for the signless Laplacian spectral radius of any general family of graphs. We have also obtained Nordhaus-Gaddum type inequality using the derived bounds.

On structural and spectral properties of reduced power graph of finite groups

The reduced power graph of a finite group [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the elements of the group [Formula: see text] and two vertices [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we first describe the structure of the reduced power graph of the finite cyclic group [Formula: see text]. Consequently, we provide a short and alternative proof of one of the results published in (R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multilinear Algebra (2019) 1–18). We characterize the values of [Formula: see text] for which [Formula: see text] is a line graph. We then deduce the signless Laplacian spectrum of [Formula: see text] using its structure. We provide lower and upper bounds on the signless Laplacian spectral radius of [Formula: see text]. Finally, we conclude the paper by determining the signless Laplacian spectrum of [Formula: see text], where [Formula: see text] denotes the dihedral group of order [Formula: see text].

On the [formula omitted]-spectral radius of graphs without linear forests

Let A(G) and D(G) be the adjacency and degree matrices of a simple graph G on n vertices, respectively. The Aα-spectral radius of G is the largest eigenvalue of Aα(G)=αD(G)+(1−α)A(G) for a real number α∈[0,1]. In this paper, for α∈(0,1), we obtain a sharp upper bound for the Aα-spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough and characterize all graphs which attain the upper bound. As a result, we completely dertermine the maximum signless Laplacian spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough.

Connected graphs of fixed order and size with maximal Aα-index: The one-dominating-vertex case

For any real number α∈[0,1], by the Aα-matrix of a graph G we mean the matrix Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. The largest eigenvalue of Aα(G) is called the Aα-index of G. Chang and Tam (2011) have proved that for every pair of integers n,k with −1≤k≤n−3, Hn,k, the graph obtained from the star K1,n−1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the Q-index (i.e., the signless Laplacian spectral radius or, equivalently, the A12-index) over all connected graphs with n vertices and n+k edges. In this paper it is proved that for every pair of integers n,k with −1≤k≤n−3, when 12<α<1 or α=12 and k≠2, the graph Hn,k is the unique connected graph that maximizes the Aα-index over all connected graphs with n vertices and n+k edges. This work extends (and also provides an alternative proof for) the above-mentioned result of Chang and Tam. A complete overview of the history of the maximal index problems is also given.

The maximum signless Laplacian spectral radius of graphs with forbidden subgraphs

The Brualdi-Solheid-Tur?n type problem has attracted wide attention recently. Let F be a set of graphs. A graph G is F -free if it does not contain any element of F as a subgraph. In this paper, when F1 = {W5,C6} and F2 = {F5}, we determine the maximum signless Laplacian spectral radius of Fi-free graph with n vertices, respectively and depict the corresponding extremal graph for i = 1,2.

The [formula omitted]-spectral radius of nonregular graphs (digraphs) and maximum degree (outdegree)

For 0≤α<1, let Aα(G)=αD(G)+(1−α)A(G) be the Aα-matrix of G, where D(G) is the diagonal degree matrix of G and A(G) is the adjacency matrix of G Let λα(G) denote the Aα-spectral radius of a graph G. Let G be a k-connected nonregular graph with n vertices, m edges, maximum degree Δ and minimum degree δ. In this paper, we show that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−1)2−(Δ−k+1)(n−k−1)+(1−α)nk2,which extends the result on the spectral radius of Xue and Liu (0000) and improves the result on the signless Laplacian spectral radius of Shiu et al. (2017). Furthermore, we also prove that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−Δ+2k−2)(n−δ−1)+k2+(1−α)nk2,which extends the result on the signless Laplacian spectral radius of Ning et al. (2018).Let Γ be a k-strong nonregular digraph of order n, size m, and maximum outdegree Δ+. For 0≤α<1, let Aα(Γ)=αD(Γ)+(1−α)A(Γ) be the Aα-matrix of Γ, where D(Γ) is the diagonal matrix of its vertex outdegrees and A(Γ) is the adjacency matrix of Γ. Denote by λα(Γ) the Aα-spectral radius of a digraph Γ. We prove that Δ+−λα(Γ)>(1−α)(nΔ+−m)k22(nΔ+−m)(n−1)2−(Δ+−k+1)(n−k−1)+(1−α)nk2,which improves the result of Xi and Wang (2020). In the end of the paper, related problem is mentioned.

On the Signless Laplacian Spectral Radius of Graphs without Small Books and Intersecting Quadrangles

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize all extremal graphs. In addition, we give an upper bound of the signless Laplacian spectral radius of all graphs which do not contain intersecting quadrangles as a subgraph.

Maxima of the Q-index: Graphs with no K1,t-minor

A graph is said to be H-minor free if it does not contain H as a minor. In spectral extremal graph theory, it is interesting to determine the maximum (signless Laplacian) spectral radius of graphs that do not contain a given H as a minor. In this paper, we characterize the unique extremal graph with the maximum signless Laplacian spectral radius among all n-vertex connected K1,t-minor free graphs (t≥3).

Sufficient spectral conditions for graphs being k-edge-Hamiltonian or k-Hamiltonian

A graph G is k-edge-Hamiltonian if any collection of vertex-disjoint paths with at most k edges altogether belong to a Hamiltonian cycle in G. A graph G is k-Hamiltonian if for all with , the subgraph induced by has a Hamiltonian cycle. These two concepts are classical extensions of the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edge-Hamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra. 2016;64:2252–2269], Nikiforov [Czechoslovak Math J. 2016;66:925–940] and Li et al. [Linear Multilinear Algebra. 2018;66:2011–2023]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which could be regarded as a complement of two recent results of Füredi et al. [Discrete Math. 2017;340:2688–2690] and [Discrete Math. 2019;342:1919–1923].

On the Aσ-spectral radii of graphs with some given parameters

Given a graph G, the adjacency matrix and degree diagonal matrix of G are denoted by A(G) and D(G), respectively. In 2017, Nikiforov proposed the Aσ-matrix: Aσ(G)=σD(G)+(1−σ)A(G), where σ∈[0,1]. The largest eigenvalue of this novel matrix is called the Aσ-index of G. Let ℬnα be the class of n-vertex block graphs with independence number α and let 𝒢(n,k) be another class of n-vertex graphs with k cut edges. We show that the maximum Aσ-index, among all graphs G∈ℬnα (resp. G∈𝒢(n,k)), is attained at a unique graph. It is surprising to see that in both cases, the extremal graphs are usually pineapple graphs. We use two methods to establish upper bounds on the Aσ-index of the corresponding extremal graphs. As a byproduct we obtain an upper bound for signless Laplacian spectral radius q1(G), when G∈ℬnα.

Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii

Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii

On Extremal Spectral Radii of Uniform Supertrees with Given Independence Number

A supertree is a connected and acyclic hypergraph. Denote by T m , n , α the set of m -uniform supertrees of order n with independent number α . Focusing on the spectral radius in T m , n , α , this present completely determines the hypergraphs with maximum spectral radius among all the supertrees with n vertices and independence number α for m − 1 / m n ≤ α ≤ n − 1 , which extend the results of Lu et al. from tree to uniform supertree. Our techniques are based on the structure properties of supertrees with given independence number and general edge-moving operation. As a byproduct, we also determine the hypergraphs with minimum signless Laplacian spectral radius in T m , n , α .

On the maximal α-spectral radius of graphs with given matching number

Let be the set of graphs of order n with given matching number β. Let be the diagonal matrix of the degrees of the graph G and be the adjacency matrix of the graph G. The largest eigenvalue of the nonnegative matrix is called the α-spectral radius of G. The graphs with maximal α-spectral radius in are completely characterized in this paper. In this way, we provide a general framework to attack the problem of extremal spectral radius in . More precisely, we generalize the known results on the maximal adjacency spectral radius in and the signless Laplacian spectral radius.

Signless Laplacian spectral radius of graphs without short cycles or long cycles

The signless Laplacian spectral radius of a graph G, denoted by q(G), is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let G(m,g) be the family of graphs on m edges with girth g and H(m,c) be the family of graphs on m edges with circumference c. More precisely, we obtain the unique extremal graph with maximal q(G) in G(m,g) and H(m,c), respectively.