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  and . However, there is not much known about digraphs.
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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  studies the bipartite graph
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 .
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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 [, ].
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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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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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 .
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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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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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).
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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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.
Motivated by  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 invariants have been used to study a variety of properties of graphs. See, for instance,    . 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.
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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 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.  Ş. B. Bozkurt, A. D. Gungor, I. Gutman and A. S. Cevik, Randić matrix and
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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 . Recently, a number of other graph energies have been introduced; for details see the recent survey . Among these, the Laplacian and Seidel energies are of importance for the following considerations.
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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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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  gave the skew Laplacian energy of a simple digraph G as
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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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