Laplacian Spectral Radius Research Articles

On the eigenvalues of the distance signless Laplacian matrix of graphs

Let G be a connected graph and let DQ(G) be the distance signless Laplacian matrix of G with eigenvalues ρ1≥ ρ2≥…≥ ρn. The spread of the matrix DQ}(G) is defined as s(DQ(G)) := maxi,j| ρi-ρj| = ρ1- ρn. We derive new bounds for the distance signless Laplacian spectral radius ρ1 of G. We establish a relationship between the distance signless Laplacian energy and the spread of DQ(G). For a real number α ≠ 0, the graph invariant mα (G) is the sum of the α -th power of the distance signless Laplacian eigenvalues of G. Finally, we obtain various bounds for the graph invariant mα(G).

Spectral sufficient conditions for graph factors containing any edge

A factor of a graph is a spanning subgraph. Spectral sufficient conditions are provided via spectral radius and signless Laplacian spectral radius for graphs with (i) a matching of given size (particularly, $1$-factor) containing any given edge, and (ii) a star factor with a component isomorphic to stars of order two or three containing any given edge, respectively.

Distance signless Laplacian spectral radius for the existence of path-factors in graphs

Distance signless Laplacian spectral radius for the existence of path-factors in graphs

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.

Hermitian skew Laplacian matrix of oriented graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] having [Formula: see text] edges and let [Formula: see text] be an orientation of [Formula: see text]. The hermitian skew Laplacian matrix [Formula: see text] of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix of oriented degrees [Formula: see text] and [Formula: see text] is the skew matrix. The spectral radius of the matrix [Formula: see text] is the hermitian skew Laplacian spectral radius and the sum of the absolute values of eigenvalues of [Formula: see text] is the hermitian skew Laplacian energy of [Formula: see text]. In this paper we study the hermitian spectral radius and hermitian skew Laplacian energy of [Formula: see text]. We obtain some new bounds for the hermitian skew Laplacian spectral radius and characterize the extremal oriented graphs attaining these bounds. Further, we obtain some bounds for the hermitian skew Laplacian energy and locate the extremal oriented graphs.

Distance Laplacian Spectral Radius of the Complements of Trees and Unicyclic Graphs

Distance Laplacian Spectral Radius of the Complements of Trees and Unicyclic Graphs

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

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

Reciprocal distance Laplacian spectral properties double stars and their complements

Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.

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).

Extremal normalized Laplacian spectral radii of graphs

The normalized Laplacian spectral radius of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graphs with the minimum normalized Laplacian spectral radius among all connected n-vertex graphs except three special graphs. As an application, the extremal graphs with the minimum normalized Laplacian spectral radius among all connected n-vertex Kr-free graphs with r≥⌈n+22⌉ are characterized completely.

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.

Laplacian eigenvalues of the unit graph of the ring [formula omitted

We investigate the Laplacian eigenvalues about unit graph G(Zn) on Zn and show the case whenever n is an even number or an odd prime power, G(Zn) would be Laplacian integral. We also prove that if n>1, then the Laplacian spectral radius of G(Zn) is equal cardinal number of V(G(Zn)) if and only if n=pk, here p is a prime integer, k is an integer that positive. In addition, this paper also characterizes n that algebraic connectivity of G(Zn) coincides with the vertex connectivity.

On the spread of the distance signless Laplacian matrix of a graph

Abstract Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1 Q(G), δ2 Q(G), ..., δn Q(G). δ1 Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.

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].

A note on the distance and distance signless Laplacian spectral radius of complements of trees

In this short note, we show that the generalized tree shift operation increases the distance spectral radius, distance signless Laplacian spectral radius, and the Dα-spectral radius of complements of trees. As a consequence of these results, we correct some ambiguities in the proofs of some known results.

On graphs with distance Laplacian eigenvalues of multiplicity n−4

Let G be a connected simple graph with n vertices. The distance Laplacian matrix D L ( G ) is defined as D L ( G ) = Diag ( T r ) − D ( G ) , where Diag ( T r ) is the diagonal matrix of vertex transmissions and D ( G ) is the distance matrix of G. The eigenvalues of D L ( G ) are the distance Laplacian eigenvalues of G and are denoted by ∂ 1 L ( G ) ≥ ∂ 2 L ( G ) ≥ ⋯ ≥ ∂ n L ( G ) . The largest eigenvalue ∂ 1 L ( G ) is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018), and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity n − 3 . In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity n − 4 together with one of the distance Laplacian eigenvalues as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity n − 4 .

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Let G be a connected graph of even order n. An odd [1, b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1, 3, 5, ⋯, b} for any v ∈ V(G), where b is positive odd integer. The distance matrix Ɗ(G) of G is a symmetric real matrix with (i, j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1, b]-factor in a graph; we provide some graphs to show that the bounds are optimal.