The Signless Laplacian Spectral Radius

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The signless Laplacian spectral radius of some strongly connected digraphs

The signless Laplacian spectral radius of some strongly connected digraphs

G ) be the diagonal matrix with out-degrees of the vertices of − → G . Then the matrix Q( − → G ) = D( − → G ) + A( − → G ) is called the signless Laplacian matrix of Q( − → G), and let q( − → G ) denote its signless Laplacian spectral radius, the largest modulus of an eigenvalue of Q( − → G ). The polynomial φ( − → G , λ) = det(λI − Q( − → G )) is defined as the characteristic polynomial with respect to the signless Laplacian matrix Q( − → G). The collection of eigenvalues of Q( − → G ) together with multipli- cates is called the Q-spectrum of − → G . There are many articles on the signless Laplacian spectrum of undirected graphs [1, 2, 12, 14, 16]. There are also many articles on the topic of adjacency spectrum [3, 6, 9], and the Laplacian spectrum [8, 13]. For addi- tional remarks on this topic we refer the reader to see two excellent surveys [10] and [11]. However, there is not much known about digraphs.

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The Signless Laplacian Spectral Radius of Some Special Bipartite Graphs

The Signless Laplacian Spectral Radius of Some Special Bipartite Graphs

DOI: 10.4236/jamp.2018.610181 2160 Journal of Applied Mathematics and Physics order of n and the size of m , and the graph with the largest Laplace spectrum ra- dius in the graph class was determined, as well as the upper bound of the Laplace spectrum radius of D m m n n ( 1 , ; , 2 1 2 ) . The actical [2] studies the bipartite graph

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On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs

On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs

get the eigenvalues of the complementary distance signless Laplacian matrix of graphs obtained by some graph operations. Finally, in the Section 6, we obtain the bounds for the complementary distance signless Laplacian energy of graphs. The results of this paper are analogous to the results obtained in [2].

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Some new sharp bounds for the spectral radius of a nonnegative matrix and its application

Some new sharp bounds for the spectral radius of a nonnegative matrix and its application

In this paper, we give some new sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. Using these bounds, we obtain some new and improved bounds for the signless Laplacian spectral radius of a graph or a digraph which are better than the bounds in [, ].

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On the Laplacian spectral characterization of $\Pi$ shape trees

On the Laplacian spectral characterization of $\Pi$ shape trees

By Lemma 32., a graph L-cospectral with Π-shape tree must be a Π-shape tree. Thus, if we can prove that no two non-isomorphic Π-shape trees are L-cospectral, then the Π-shape tree is determined by its Laplacian spectrum. Unfortunately, we find that there exist many L-cospectral mates in Π-shape tree. For convenience we classify the Π-shape trees into two types according to l 2 , i.e., Π =0 and Π ≥1 .

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Some new estimations for the upper and lower bounds for the spectral radius of nonnegative matrices

Some new estimations for the upper and lower bounds for the spectral radius of nonnegative matrices

But in fact, ρ(A ◦ B) = .. So inequality in () does not hold in this example. Thus the upper bound in () does not apply for all spectral radius of the Hadamard product of non- negative matrices. Since the proof in Theorem . in [], i.e., mainly applied Lemma ., and r li <  (l = i) when |a ll | –

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Laplacian Spectral Characterization of Signed Sun Graphs

Laplacian Spectral Characterization of Signed Sun Graphs

Spectral characterizations of certain classes of graphs is an important subject in spectral graph theory. This means if a graph G in the specified class is cospectral with a graph H (not necessarily in the class), then G and H are isomorphic. This study is extended to signed graphs by some authors [1].

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Conformal variations of the spectral zeta function of the Laplacian

Conformal variations of the spectral zeta function of the Laplacian

Consequently, it becomes of much interest to study how the Casimir energy and other points of the spectral zeta function vary under more general deformation of the metric such as conformal perturbation of the Riemannian manifold, and in fact, it is sensible to fix the volume so as to factor out this trivial scaling.

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A new bound for the spectral radius of nonnegative tensors

A new bound for the spectral radius of nonnegative tensors

By estimating the ratio of the smallest component and the largest component of a Perron vector, we provide a new bound for the spectral radius of a nonnegative tensor. And it is proved that the proposed result improves the bound in (Li and Ng in Numer. Math. 130(2):315-335, 2015).

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New bounds for the spectral radius for nonnegative tensors

New bounds for the spectral radius for nonnegative tensors

A lower bound and an upper bound for the spectral radius for nonnegative tensors are obtained. A numerical example is given to show that the new bounds are sharper than the corresponding bounds obtained by Yang and Yang (SIAM J. Matrix Anal. Appl. 31:2517-2530, 2010), and that the upper bound is sharper than that obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014).

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Joint spectral radius, Sturmian measures and the finiteness conjecture

Joint spectral radius, Sturmian measures and the finiteness conjecture

Sturmian intervals in Section 7, the key Section 8 establishes the link between Stur- mian intervals and the parameter t of the pair A(t). Section 9 treats the case of those parameters t such that one matrix in the pair A(t) dominates the other, so that the joint spectral radius r(A(t)) is simply the spectral radius of the dominating matrix. All other parameters are considered in Section 10, establishing that the joint spectral radius is always attained by a unique Sturmian measure. Finally, in Section 11 we show that the map taking parameter values t to the associated Sturmian parameter P (t) is a devil’s staircase.

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On the spectral radius of bipartite graphs which are nearly complete

On the spectral radius of bipartite graphs which are nearly complete

In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [–]. As a special case, in [], Chen et al. studied the spectral radius of bipartite graphs which are close to a complete bipartite graph. For partite sets U and V having |U| = p, |V | = q and p ≤ q, in the same reference, the authors also gave an affirmative answer to the conjecture [, Conjecture .] by taking | E(G) | = pq –  into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least pq – p + , there still exists the following conjecture.

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Some inequalities on the spectral radius of matrices

Some inequalities on the spectral radius of matrices

Motivated by [8] and [1–4, 9, 10], in this paper we propose some inequalities on the up- per bounds for the spectral radius of the Hadamard product of any k nonnegative matri- ces. These bounds generalize some existing results, and some comparisons between these bounds are also considered.

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Spectral Radius and Some Hamiltonian Properties of Graphs

Spectral Radius and Some Hamiltonian Properties of Graphs

Spectral invariants have been used to study a variety of properties of graphs. See, for instance, [9] [5] [6] [7]. In this note, we will present sufficient conditions which are based on the spectral radius for some Hamiltonian properties of graphs. The results are as follows.

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The Signless Laplacian Estrada Index of Unicyclic Graphs

The Signless Laplacian Estrada Index of Unicyclic Graphs

In this paper, we have determined the unicyclic graphs with the first two largest and smallest SLEE’s. We have also specified the unique graph with maximum SLEE among all unicyclic graphs on n vertices with a given diameter. Indeed, the main idea of this paper (also [9, 12, 13]), is to use the notion of the signless Laplacian spectral moments of graphs to compare their SLEE’s.

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Note on the Sum of Powers of Normalized Signless Laplacian Eigenvalues of Graphs

Note on the Sum of Powers of Normalized Signless Laplacian Eigenvalues of Graphs

[1] M. Bianchi, A. Cornaro, J. L. Palacios and A. Torriero, Bounding the sum of powers of normalized Laplacian eigenvalues of graphs through majorization methods, MATCH Commun. Math. Comput. Chem. 70 (2013) 707–716. [2] Ş. B. Bozkurt, A. D. Gungor, I. Gutman and A. S. Cevik, Randić matrix and

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Seidel Signless Laplacian Energy of Graphs

Seidel Signless Laplacian Energy of Graphs

The spectral graph theory based on the eigenvalues of the adjacency matrix is the most extensively elaborated part of algebraic graph theory [4, 6]. Also the graph energy, based on the eigenvalues of the adjacency matrix, attracted much attention and has been studied to a great extent [19]. Recently, a number of other graph energies have been introduced; for details see the recent survey [16]. Among these, the Laplacian and Seidel energies are of importance for the following considerations.

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Survey of Graph Energies

Survey of Graph Energies

The graph energy is a graph–spectrum–based quantity, introduced in the 1970s. After a latent period of 20–30 years, it became a popular topic of research both in mathematical chemistry and in “pure” spectral graph theory, resulting in over 600 published papers. Eventually, scores of different graph energies have been conceived. In this article we provide the basic facts on graph energies, in particular historical and bibliographic data.

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Bounds for the signless Laplacian energy of digraphs

Bounds for the signless Laplacian energy of digraphs

Since the skew Laplacian matrix of a digraph contains the information of the degree of vertices and it is a skew symmetric matrix. It plays an important role in spectral theory of digraphs. Therefore, based on the definition of the Laplacian energy of undirected graphs, Adiga and Khoshbakht [2] gave the skew Laplacian energy of a simple digraph G as

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On the Laplacian spectral radii of Halin graphs

On the Laplacian spectral radii of Halin graphs

Let T be a tree with at least four vertices, none of which has degree 2, embedded in the plane. A Halin graph is a plane graph constructed by connecting the leaves of T into a cycle. Thus the cycle C forms the outer face of the Halin graph, with the tree inside it. Let G be a Halin graph with order n. Denote by μ (G) the Laplacian spectral radius of G. This paper determines all the Halin graphs with μ (G) ≥ n – 4. Moreover, we obtain the graphs with the first three largest Laplacian spectral radius among all the Halin graphs on n vertices.

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