Signless Laplacian Eigenvalues Research Articles

Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The signless Laplacian matrix of a graph G is defined as Q(G)=D(G)+A(G), where D(G) is the degree-diagonal matrix of G and A(G) is the adjacency matrix of G. The multiplicity of an eigenvalue μ of Q(G) is denoted by mQ(G,μ). Let G=C(T1,…,Tg) be a unicyclic graph with a perfect matching, where Ti is a rooted tree attached at the vertex vi of the cycle Cg. It is proved that Q(G) has 2 as an eigenvalue if and only if g+t is divisible by 4, where t is the number of Ti of even orders. Another main result of this article gives a characterization for a connected graph G with a perfect matching such that mQ(G,2)=θ(G)+1, where θ(G)=|E(G)|−|V(G)|+1 is the cyclomatic number of G.

Relationship Between the Second Largest Adjacency and Signless Laplacian Eigenvalues of Graphs and Properties of Planar Graphs

Relationship Between the Second Largest Adjacency and Signless Laplacian Eigenvalues of Graphs and Properties of Planar Graphs

Distribution of signless Laplacian eigenvalues and graph invariants

For a simple graph on n vertices, any of its signless Laplacian eigenvalues is in the interval [0,2n−2]. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in [0,2n−2] and graph invariants including matching number and diameter.

On the sum of the first two largest signless Laplacian eigenvalues of a graph

For a graph G, let S2(G) be the sum of the first two largest signless Laplacian eigenvalues of G, and f(G)=e(G)+3−S2(G). Oliveira, Lima, Rama and Carvalho conjectured that K1,n−1+ (the star graph with an additional edge) is the unique graph with minimum value of f(G) on n vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of S2(G) proposed by Ashraf et al.

Eigenvalues of the generalized subdivision graph with applications to graph energy

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], let [Formula: see text] denotes the subdivision graph of [Formula: see text] with vertex set [Formula: see text]. In [Formula: see text], replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. Then in the resulting graph, replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. The resulting graph is denoted by [Formula: see text]. This generalizes the construction of the subdivision graph [Formula: see text] to [Formula: see text] of a graph [Formula: see text]. In this paper, we provide the complete information about the spectrum of [Formula: see text] using the spectrum of [Formula: see text]. Further, we determine the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text], when [Formula: see text] is a regular graph. Also, we find the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text] when [Formula: see text]. The energy of a graph [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of [Formula: see text]. The incidence energy of a graph [Formula: see text] is defined as the sum of the square roots of the signless Laplacian eigenvalues of [Formula: see text]. Finally, as an application, we show that the energy of the graph [Formula: see text] is completely determined by the incidence energy of the graph [Formula: see text].

Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛

Abstract Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

Signless Laplacian spectrum of a graph

This paper presents tight upper bounds for all of the signless Laplacian eigenvalues of a graph with prescribed order and minimum degree, which improve previously known upper bounds. Also, the relation between the number of signless Laplacian eigenvalues falling within specific intervals and various graph parameters such as independence, clique, chromatic, edge covering and matching numbers are explored.

Spectrum of the Cozero-Divisor Graph Associated to Ring Zn

Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)′, w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′(Zn). We also show that the cozero-divisor graph Γ′(Zp1p2) is a signless Laplacian integral.

Some new inequalities on the normalized signless Laplacian resolvent energy of graphs

For a connected graph [Formula: see text] of order [Formula: see text] with normalized signless Laplacian eigenvalues [Formula: see text], the normalized signless Laplacian resolvent energy of [Formula: see text] is defined as [Formula: see text]. In this paper, we derive new inequalities on [Formula: see text] as well as relations on [Formula: see text] with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature.

(Generalized) Incidence and Laplacian-Like Energies

In this study, for graph Γ with r connected components (also for connected nonbipartite and connected bipartite graphs) and a real number ε ≠ 0,1 , we found generalized and improved bounds for the sum of ε -th powers of Laplacian and signless Laplacian eigenvalues of Γ . Consequently, we also generalized and improved results on incidence energy I E and Laplacian energy-like invariant L E L .

Bounding the sum of the largest signless Laplacian eigenvalues of a graph

We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.

On graphs with eigenvectors in {−1,0,1} and the max k-cut problem

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to {−1,0,1}. We extend the graph parameter max k-cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max k-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite families of extremal graphs for the obtained upper bounds.

Remarks on the sum of powers of normalized signless Laplacian eigenvalues of graphs

Let G = (V, E), V = {v1, v2,..., vn}, be a simple connected graph of order n and size m. Denote by ?+1 ? ?+2 ?...? ?+n ? 0 the normalized signless Laplacian eigenvalues of G, and by ??(G) the sum of ?-th powers of the normalized signless Laplacian eigenvalues of a connected graph. The paper deals with bounds of ??. Some special cases, when ? = 12 and ? = ?1, are also considered.

Bounds on some energy-like invariants of corona and edge corona of graphs

The Laplacian-energy-like invariant of a finite simple graph is the sum of square roots of all its Laplacian eigenvalues and the incidence energy is the sum of square roots of all its signless Laplacian eigenvalues. In this paper, we give the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona of two graphs. We also observe that the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona are sharp when the graph is the corona or edge corona of two complete graphs.

On the multiplicity of the least signless Laplacian eigenvalue of a graph

Let G be a simple and connected graph of order n(≥6). The eigenvalues of the signless Laplacian matrix Q(G) of G are called the Q-eigenvalues of G. Let qn(G) be the least Q-eigenvalue of G. In this paper, we investigate the multiplicity of qn(G) and characterize all n-vertex connected graphs with qn(G) of multiplicity n−3. Moreover, it is shown that all these graphs are determined by their signless Laplacian spectra.

Signless Laplacian state transfer on [formula omitted]-graphs

For a simple graph G, its Q-graph Q(G) is derived from G by adding one new point in every edge of G and linking two new vertices by edge if they are between two edges that having a common endpoint. In our work, we demonstrate that for a regular graph G, if all the signless Laplacian eigenvalues are integers, then the Q(G) exists no signless Laplacian perfect state transfer. We also present a sufficient restriction that the Q(G) admits signless Laplacian pretty good state transfer when G exhibits signless Laplacian perfect state transfer between two specific vertices for a regular graph G. In addition, in view of these results, we also present some new families of Q-graphs, which have no signless Laplacian perfect state transfer, but admit signless Laplacian pretty good state transfer.

On the smallest signless Laplacian eigenvalue of graphs

For a simple graph G the signless Laplacian matrix of G is defined as D(G)+A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of the signless Laplacian matrix of G. In this paper we study the smallest signless Laplacian eigenvalue of graphs and find some relations between this and the chromatic number of graphs. We prove that if G is a graph of order n and with chromatic number χ(G), then q′(G)≤q′(T(n,χ(G))), where T(n,t) is the Turán graph on n vertices and t parts. Using this inequality we obtain some bounds for q′(G) that improve the known previous bounds.

Graphs with the second signless Laplacian eigenvalue ≤ 4

AbstractWe discuss the question of classifying the connected simple graphsHfor which the second largest eigenvalue of the signless LaplacianQ(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo <img src=image/13424080_01.gif>

Let <img src=image/13424080_02.gif> be a simple graph of order <img src=image/13424080_03.gif> and let <img src=image/13424080_04.gif> be the Seidel matrix of <img src=image/13424080_02.gif>, defined as <img src=image/13424080_05.gif> where <img src=image/13424080_06.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are adjacent and <img src=image/13424080_09.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are not adjacent and <img src=image/13424080_10.gif> if <img src=image/13424080_11.gif>. Let <img src=image/13424080_12.gif> be the diagonal matrix where <img src=image/13424080_13.gif> denotes the degree of the <img src=image/13424080_14.gif> vertex of <img src=image/13424080_02.gif>. The Seidel Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_15.gif> and the Seidel signless Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_16.gif>. The zero-divisor graph of a commutative ring <img src=image/13424080_17.gif>, denoted by <img src=image/13424080_18.gif>, is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices <img src=image/13424080_19.gif> are adjacent if and only if <img src=image/13424080_20.gif>. In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on <img src=image/13424080_21.gif>. Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of <img src=image/13424080_21.gif> for some values of <img src=image/13424080_03.gif>, say <img src=image/13424080_22.gif>, where <img src=image/13424080_23.gif> are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of <img src=image/13424080_24.gif>, for any <img src=image/13424080_03.gif>.

Minimum values of the second largest [formula omitted]-eigenvalue

For a graph G, the signless Laplacian matrix Q(G) is defined as Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) the diagonal matrix whose main entries are the degrees of the vertices in G. The Q-spectrum of G is that of Q(G). In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q2(G) of a connected graph G. We find the five smallest values of q2(G) over the set of connected graphs G with given order n. We also characterize the corresponding extremal graphs.