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.

15 Read more

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

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].

21 Read more

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 .

15 Read more

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 | –

14 Read more

**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].

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.

23 Read more

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).

12 Read more

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.

38 Read more

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 [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.

12 Read more

**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.

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.

13 Read more

[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

12 Read more

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.

11 Read more

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.

45 Read more

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

11 Read more

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.

18 Read more