Search

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 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]

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]

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]

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]

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]

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]