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The normalized distance Laplacian of a graph G is defined as D L (G) = T (G) − 1 / 2 (T (G) − D (G)) T (G) − 1 / 2 where D (G) is the matrix with pairwise distances between vertices and T (G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius. [ABSTRACT FROM AUTHOR]

The Maxwell-Smoluchowski (MS) theory of gas diffusion is revisited here in the context of gas transport in straight channels in the Knudsen regime of large mean free path. This classical theory is based on a phenomenological model of gas-surface interaction that posits that a fraction ϑ of molecular collisions with the channel surface consists of diffuse collisions, i.e., the direction of post-collision velocities is distributed according to the Knudsen Cosine Law, and a fraction 1 − ϑ undergoes specular reflection. From this assumption one obtains the value D = 2 − ϑ ϑ D K for the self-diffusivity constant, where D K is a reference value corresponding to ϑ = 1. In this paper we show that ϑ can be expressed in terms of micro- and macro-geometric parameters for a model consisting of hard spheres colliding elastically against a rigid surface with prescribed microgeometry. Our refinement of the MS theory is based on the observation that the classical surface scattering operator associated to the microgeometry has a canonical velocity space diffusion approximation by a generalized Legendre differential operator whose spectral theory is known explicitly. More specifically, starting from an explicit description of the effective channel surface microgeometry—a concept which incorporates both the actual surface microgeometry and the molecular radius—and using this operator approximation, we show that ϑ can be resolved into easily obtained geometric parameters, ϑ = λ h / C , having the following interpretation: C is a macroscopic parameter determined by the shape of the channel cross-section; h is a parameter that precisely captures the degree of roughness of the effective microgeometry, and λ is a parameter that characterizes the overall curvature of the surface microgeometry independent of h. Thus ϑ is resolved as the quotient of microscopic (λh) over macroscopic (C) signature parameters of the channel geometry. The identity ϑ = λ h / C holds up to higher order terms in the roughness parameter h , so our main result better applies to well polished, or low roughness, surfaces. • The Maxwell-Smoluchowski theory of gas diffusion is revisited in the context of gas transport in straight channels in the Knudsen regime of large mean free path, in the special case of channel surface having low roughness. • A factorization of the phenomenological tangential momentum accommodation coefficient introduced in the Maxwell-Smoluchowski model is derived in terms of micro- and macro-geometric parameters based on a detailed analysis of the surface scattering of gas molecules under the assumption of low roughness. • A new concept of roughness is proposed which enters into the description of self-diffusivity in a more direct way than standard definitions, and these different notions of roughness are compared. • Utilizing a very simple model of micro-geometry, the new analysis obtains estimates for self-diffusivity of gas transport in carbon nanotubes in line with the very high values observed experimentally. [ABSTRACT FROM AUTHOR]

Let D (G) and D Q (G) = D i a g (T r) + D (G) be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph G , respectively, where D i a g (T r) = diag (D 1 , D 2 , ... , D n) be the diagonal matrix with vertex transmissions of the digraph G. To track the gradual change of D (G) into D Q (G) , in this paper, we propose to study the convex combinations of D (G) and D i a g (T r) defined by D α (G) = α D i a g (T r) + (1 − α) D (G) , 0 ≤ α ≤ 1. This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of D α (G) is called the D α spectral radius of G , denoted by μ α (G). We determine the digraph which attains the maximum (or minimum) D α spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum D α spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to give a new formula of the joint spectral radius of commuting d -tuples of operators on a complex Hilbert space. Also we show that r ( T ) ≤ ω ( T ) for a commuting d -tuple of operators T , where r ( T ) and ω ( T ) denote respectively the joint spectral radius and the joint numerical radius of T . This generalizes the well known relation between the spectral and the numerical radii of Hilbert space operators which is proved in [10] . [ABSTRACT FROM AUTHOR]

Let Ψ 1 , … , Ψ m be bounded sets of positive kernel operators on a Banach function space L . We prove that for the generalized spectral radius ρ and the joint spectral radius ρ ˆ the inequalities ρ ( Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) ) ≤ ρ ( Ψ 1 Ψ 2 ⋯ Ψ m ) 1 m , ρ ˆ ( Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) ) ≤ ρ ˆ ( Ψ 1 Ψ 2 ⋯ Ψ m ) 1 m hold, where Ψ 1 ( 1 m ) ∘ ⋯ ∘ Ψ m ( 1 m ) denotes the Hadamard (Schur) geometric mean of the sets Ψ 1 , … , Ψ m . [ABSTRACT FROM AUTHOR]

The unique graphs with maximum spectral radius and signless Laplacian spectral radius are determined among trees, unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices (vertices of degree at least 3) using a unified approach. [ABSTRACT FROM AUTHOR]

By a unified approach, we present sufficient conditions based on spectral radius for a graph to be k -connected, k -edge-connected, k -Hamiltonian, k -edge-Hamiltonian, β -deficient and k -path-coverable. [ABSTRACT FROM AUTHOR]

Let G be a simple graph, and let G σ be an oriented graph of G with skew adjacency matrix S ( G σ ) . The skew spectral radius ρ s ( G σ ) of G σ is defined as the spectral radius of S ( G σ ) . When G is an odd-cycle graph (no even cycle), Cavers et al. (2012) [4] showed that the skew spectral radius of G σ is the same for every orientation σ of G . They proposed a problem: If G is a connected graph and ρ s ( G σ ) is the same for all orientations σ of G , must G be an odd-cycle graph? In this paper, we solve this problem and give a positive answer. [ABSTRACT FROM AUTHOR]

The lower spectral radius of a set of d × d matrices is defined to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a finite set of matrices of fixed cardinality it is known that the lower spectral radius can vary discontinuously as a function of the matrix entries. In a previous article the author and J. Bochi conjectured that when considered as a function on the set of all pairs of 2 × 2 real matrices, the lower spectral radius is discontinuous on a set of positive (eight-dimensional) Lebesgue measure, and related this result to an earlier conjecture of Bochi and Fayad. In this article we investigate the continuity of the lower spectral radius in a simplified context in which one of the two matrices is assumed to be of rank one. We show in particular that the set of discontinuities of the lower spectral radius on the set of pairs of 2 × 2 real matrices has positive seven-dimensional Lebesgue measure, and that among the pairs of matrices studied, the finiteness property for the lower spectral radius is true on a set of full Lebesgue measure but false on a residual set. [ABSTRACT FROM AUTHOR]

Let ρ ( D ( G )) denote the distance spectral radius of a graph G and ∂ ( G → ) denote the distance signless Laplacian spectral radius of a digraph G → . Let G n , k D be the set of all k -connected graphs of order n with diameter D . In this paper, we first determine the unique graph with minimum distance spectral radius in G n , k D ; we then give sharp upper and lower bounds for the distance signless Laplacian spectral radius of strongly connected digraphs; we also characterize the digraphs having the maximal and minimal distance signless Laplacian spectral radii among all strongly connected digraphs; furthermore, we determine the extremal digraph with the minimal distance signless Laplacian spectral radius with given dichromatic number. [ABSTRACT FROM AUTHOR]

For a connected graph G , the distance signless Laplacian spectral radius of G is the spectral radius of its distance signless Laplacian matrix Q ( G ) defined as Q ( G ) = T r ( G ) + D ( G ) , where T r ( G ) is the diagonal matrix of vertex transmissions of G and D ( G ) is the distance matrix of G . In this paper, we study the effect of three types of graft transformations to decrease and/or increase the distance signless Laplacian spectral radius. As applications, on one hand, we determine the unique graphs with minimum distance signless Laplacian spectral radius among non-starlike trees, among non-caterpillar trees, and among non-starlike non-caterpillar trees, respectively, and determine the unique trees with third and forth minimum distance signless Laplacian spectral radius, respectively, and on the other hand, we determine the unique graphs with maximum distance signless Laplacian spectral radius among trees with given maximum degree, among trees of a perfect matching with maximum degree, among non-starlike trees, among non-caterpillar trees, among non-starlike non-caterpillar trees, and among unicyclic odd-cycle graphs, respectively, and determine the unique trees with second and third maximum distance signless Laplacian spectral radius, respectively, and the unique connected graph with second maximum distance signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

Let D ( G ) be the distance matrix of a connected graph G . The distance spectral radius of G , denoted by ∂ 1 ( G ) , is the largest eigenvalue of D ( G ) . In this paper we present a new transformation of a certain graph G that decreases ∂ 1 ( G ) . With the transformation, we partially confirm a conjecture proposed by Stevanović and Ilić [17] by showing that, if Δ ≥ ⌈ n 2 ⌉ , the double star S Δ , n − Δ uniquely minimizes the distance spectral radius among all trees on n vertices with maximum degree Δ . Moreover, the trees on n ≥ 10 vertices with the fourth and fifth least distance spectral radii are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition on the spectral radius for a graph to be Hamiltonian. [ABSTRACT FROM AUTHOR]

In the paper, we will determine the graphs with maximal spectral radius among all the trees with n vertices and independence number α for ⌈ n 2 ⌉ ≤ α ≤ n − 1 . [ABSTRACT FROM AUTHOR]

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanović (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D . [ABSTRACT FROM AUTHOR]

For a non-negative matrix A the spectral radius of the product XA is maximized over all non-negative diagonal matrices X with trace 1. Instead of following the naive approach of solving a sequence of matrix eigenvalue problems, we construct a related minimization problem, with a rather simple gradient flow, and follow this flow with a steepest descent method. This procedure gives lower bounds and eventually the solution with desired accuracy. On the other hand, we obtain an upper bound in the form of the max algebra Perron root of the matrix A (and some refined upper bounds). Numerical experiments show that in many cases the upper bound is a surprisingly good estimate. [ABSTRACT FROM AUTHOR]

We study the effect of a graft transformation on the distance spectral radius of connected graphs. As applications, we determine the unique n-vertex non-starlike tree with maximal distance spectral radius, and the unique n-vertex non-caterpillar trees with minimal and maximal distance spectral radii, and we also consider the distance spectral radius of non-starlike non-caterpillar trees. [ABSTRACT FROM AUTHOR]

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag (D) be the diagonal matrix with outdegrees of the vertices of D and Q(D)=diag (D)+A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give a sharp bound on q(D) where D has a given outdegree sequence and compare the bound with known bounds. We establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the extremal graph. In addition, we also determine the unique digraph which achieves the minimum (or maximum), the second minimum (or maximum), the third minimum, the fourth minimum spectral radius and signless Laplacian spectral radius among all strongly connected digraphs, and answer the open problem proposed by Lin and Shu [14] [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple connected weighted graph on n vertices, in which the edge weights are positive definite matrices. The eigenvalues of G are the eigenvalues of its adjacency matrix. In this note, we present a correction in equality part in Theorem 2 [S. Sorgun, S. Büyükköse, The new upper bounds on the spectral radius of weighted graphs, Appl. Math. Comput. 218 (2012) 5231–5238]. In addition, some related results are also provided. [ABSTRACT FROM AUTHOR]

Abstract: We show that the so-called diamond alternating sign matrix is the unique alternating sign matrix with maximum spectral radius , and that . [Copyright &y& Elsevier]

Abstract: Let G be a simple connected graph of order n with degree sequence in non-increasing order. The signless Laplacian spectral radius of G is the largest eigenvalue of its signless Laplacian matrix . In this paper, we give a sharp upper bound on the signless Laplacian spectral radius in terms of , which improves and generalizes some known results. [Copyright &y& Elsevier]

Abstract: The distance spectral radius of a graph G is the largest eigenvalue of the distance matrix . In this paper, we characterize the graph with minimum distance spectral radius among trees with fixed number of pendent vertices. [Copyright &y& Elsevier]

Abstract: The spectral radius of a graph G is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph G with is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radii greater than . In this paper we proved the following results. For any open quipu G on n vertices () with spectral radius less than , its diameter satisfies . This bound is tight. For any closed quipu G on n vertices () with spectral radius less than , its diameter satisfies . The upper bound is tight while the lower bound is asymptotically tight. Let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D. For and , we proved that is the unique graph obtained by attaching two paths of lengths and to a pair of antipodal vertices of the even cycle . This result is tight. Thus we settled a conjecture of Cioabaˇ–van Dam–Koolen–Lee, who previously proved a special case for and n large enough. [Copyright &y& Elsevier]

Abstract: In this article, we determine the unique graph with maximal distance spectral radius in the class of graphs without a pendent vertex. [Copyright &y& Elsevier]

Abstract: In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [8]. [Copyright &y& Elsevier]

Abstract: In this paper, we present preconditioned generalized accelerated overrelaxation methods. We compare the spectral radii of the iteration matrices of the preconditioned and original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we present two numerical examples to confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: Let be a simple connected graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . In this paper, we obtain some new and improved sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph G. [Copyright &y& Elsevier]

Abstract: In this paper, we determine the unique graph with minimum distance spectral radius among all connected bipartite graphs of order with a given matching number. Moreover, we characterize the graphs with minimal distance spectral radius in the class of all connected bipartite graphs with a given vertex connectivity. [Copyright &y& Elsevier]

Abstract: Let denote the distance matrix of a connected graph with order , where is equal to the distance between and in . The largest eigenvalue of is called the distance spectral radius of graph , denoted by . In this paper, some graft transformations that decrease or increase are given. With them, for the graphs with both order and pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when ; for , the extremal graphs with the maximum distance spectral radius are completely characterized. [Copyright &y& Elsevier]

Abstract: If G is a connected undirected simple graph on n vertices and edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if . Let be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices. [Copyright &y& Elsevier]

Abstract: The eigenvalue problem for a linear function centers on solving the eigen-equation . This paper generalizes the eigenvalue problem from a single linear function to an iterated function system consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes , where is real, is a compact set, and . The main result is that an irreducible, linear iterated function system has a unique eigenvalue equal to the joint spectral radius of the functions in and a corresponding eigenset that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranisnikov–Konyagin–Protasov on the joint spectral radius follow as corollaries. [Copyright &y& Elsevier]

Abstract: The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we provide structural and behavioral details of graphs with maximum Laplacian spectral radius among all bipartite connected graphs of given order and size. Using these results, we provide a unified approach to determine the graphs with maximum Laplacian spectral radii among all trees, and all bipartite unicyclic, bicyclic, tricyclic and quasi-tree graphs, respectively. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph with vertices , of degrees , respectively. Let A be the -adjacency matrix of G and D be the diagonal matrix diag. is called the signless Laplacian of G. The largest eigenvalue of is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number and the minimal Q-spectral radius are characterized as well. [Copyright &y& Elsevier]

Abstract: Let be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized. [Copyright &y& Elsevier]

Abstract: Denote by the set of n-vertex weighted trees with matching number q and fixed positive weight set , where . Tan [S.W. Tan, On the sharp upper bound of spectral radius of weighted trees, J. Math. Res. Exposition 29 (2009) 293–301] determined the weighted tree in with the largest adjacent spectral radius , whereas in [S.W. Tan, On the Laplacian spectral radius of weighted trees with a positive weight set, Discrete Math. 310 (2010) 1026–1036] Tan determined the weighted tree in with the largest Laplacian spectral radius. In this paper, we use a unified approach to identify the unique weighted tree in with the largest adjacent spectral radius and largest Laplacian spectral radius, respectively. [Copyright &y& Elsevier]

Abstract: In this work, we introduce the class of -matrices for the max algebra and derive some properties of these that echo similar results for -matrices in the conventional algebra. In analogy with [1], we define the row--property and the S max -property of a finite set of nonnegative matrices. Moreover, we relate these concepts to stability questions for sets of matrices and difference inclusions in the max algebra. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices ( fixed), which achieves the maximal signless Laplacian spectral radius? We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at , where is a tree obtained from a star and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of and give some results. [Copyright &y& Elsevier]

Abstract: Given a bounded set of non-negative matrices, let and denote the generalized spectral radius of and its max version, respectively. We show thatwhere denotes the Hadamard power of . We apply this result to give a new short proof of a known fact that is continuous on the Hausdorff metric space of all nonempty compact collections of non-negative matrices. [Copyright &y& Elsevier]

Abstract: In this paper, we give some results on Laplacian spectral radius of graphs with cut vertices, and as their applications, we also determine the unique graph with the largest Laplacian spectral radius among all unicyclic graphs with vertices and diameter , . [ABSTRACT FROM AUTHOR]

Abstract: A connected graph is called a quasi-k-cyclic graph, if there exists a vertex such that is a k-cyclic graph (connected with cyclomatic number k). In this paper we identify in the set of quasi-k-cyclic graphs (for ) those graphs whose spectral radius of the adjacency matrix (and the signless Laplacian if ) is the largest. In addition, for quasi-unicyclic graphs we identify as well those graphs whose spectral radius of the adjacency matrix is the second largest. [Copyright &y& Elsevier]

Abstract: The independence number of is defined as the maximum cardinality of a set of pairwise non-adjacent vertices which is called an independent set. In this paper, we characterize the graphs which have the minimum spectral radius among all the connected graphs of order with independence number . [Copyright &y& Elsevier]

Abstract: A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. [Copyright &y& Elsevier]

Abstract: We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence. [Copyright &y& Elsevier]

Abstract: Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and be the largest eigenvalue of the adjacency matrix of . We study how large can be when does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems. [Copyright &y& Elsevier]

Abstract: In this primarily expository paper we survey classical and some more recent results on the spectra of digraphs, equivalently, the spectra of -matrices, with emphasis on the spectral radius. [Copyright &y& Elsevier]

Abstract: We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide. [Copyright &y& Elsevier]

Abstract: This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product. Here we address the problem of establishing the finiteness property of pairs of sign-matrices. Such problem is related to the conjecture that pairs of sign-matrices fulfil the finiteness property for any dimension. This would imply, by a recent result by Blondel and Jungers, that finite sets of rational matrices fulfil the finiteness property, which would be very important in terms of the computation of the joint spectral radius. The technique used in this paper could suggest an extension of the analysis to sign-matrices, which still remains an open problem. As a main tool of our proof we make use of a procedure to find a so-called real extremal polytope norm for the set. In particular, we present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. For pairs of sign-matrices we develop the computations exactly and hence are able to prove analytically the finiteness property. On the other hand, the algorithm can be used in a floating point arithmetic and provide a general tool for approximating the joint spectral radius of a set of matrices. [Copyright &y& Elsevier]

Abstract: We study graphs with spectral radius at most and refine results by Woo and Neumaier [R. Woo, A. Neumaier, On graphs whose spectral radius is bounded by , Graphs Combinatorics 23 (2007) 713–726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E. R. Van Dam, R. E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419] and show that the conjecture is false for the other instances. [Copyright &y& Elsevier]

Abstract: We give tight conditions on the signless Laplacian spectral radius of a graph for the existence of Hamiltonian paths and cycles. [Copyright &y& Elsevier]