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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]

In this work, we give several new upper and lower bounds for the p-numerical radii of operators as well as for n × n operator matrices. An application to 2-nilpotent operators is provided, and a p-numerical radius power inequality is also given. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , the generalized distance matrix D α (G) is defined as D α (G) = α Tr (G) + (1 − α) D (G) , 0 ≤ α ≤ 1 , where Tr (G) and D (G) are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of G. In this paper, we will study the spectral radius of the generalized distance matrix D α (G) of a connected graph G. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this note is to present upper bounds estimations for the numerical radius of a products and Hadamard products of special matrices, including sectorial and accretivedissipative matrices. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 1, a k-distance enclaveless number (or k-distance B-differential) of a connected graph G = (V, E) is Ψ k (G) = max{|(V − X) ∩ Nk,G(X)| : X ⊆ V }. In this paper, we establish upper bounds on the k-distance enclaveless number of a graph in terms of its diameter, radius and girth. Also, we prove that for connected graphs G and H with orders n and m respectively, Ψk (G × H) ≤ mn − n − m + Ψk (G) + Ψk (H) + 1, where G × H denotes the direct product of G and H. In the end of this paper, we show that the k-distance enclaveless number Ψk (T) of a tree T on n ≥ k + 1 vertices and with n1 leaves satisfies inequality Ψ k (T) ≤ k(2n−2+n1) 2k+1 and we characterize the extremal trees. [ABSTRACT FROM AUTHOR]

For a hypergraph H, let B(H) be its incidence matrix. The signless Laplacian matrix Q (H) = B (H) B (H) T , whose (i, j)-element is exactly the number of edges containing vertices v i and v j for i ≠ j and (i, i)-element is exactly the degree of vertex v i . The incidence Q-tensor Q ∗ = B (H) I B (H) T , whose (i 1 , i 2 , ... , i k) -entry of Q ∗ is exactly equal to the number of edges e of H satisfying i t ∈ e for all t ∈ [ k ] . Obviously, we can see more complete structural properties of H from Q ∗ than Q(H). Up to now, few tensors whose entries are directly related to structural properties of the corresponding hypergraphs. In this regard, we believe that it deserve to study the structural properties of hypergraphs though this tensor. In this paper, we will study some upper bounds on the spectral radius of Q ∗ by some parameters on hypergraphs. [ABSTRACT FROM AUTHOR]

In this article, we present many new bounds for the numerical radius for 2x2 block matrices involving accretive-dissipative matrices. These new bounds are represented using the spectral norm. [ABSTRACT FROM AUTHOR]

Let G n , k D be the set of all k -connected graphs of order n with diameter D. In this paper, we determine the graphs with maximal signless Laplacian spectral radius among all graphs in G n , k D. [ABSTRACT FROM AUTHOR]

Let G be a connected irregular graph on n vertices with maximum degree Δ and diameter D. The spectral radius of G, which is denoted by ρ (G) , is the largest eigenvalue of the adjacency matrix of G. In this paper, we study the lower bound of Δ - ρ (G) . As a result, a new lower bound is obtained which improves the known lower bounds of Δ - ρ (G) . [ABSTRACT FROM AUTHOR]

The spectral radius ρ (G) of a graph G is the largest eigenvalue of its adjacency matrix A (G). In this paper, we first give a sharp lower bound of the spectral radius of a graph G in terms of the degree and the average 2-degree which generalizes a result of Das and Kumar (2003), and then as an application, a sharp upper bound of the energy of G is given which generalizes a result of Liu et al. (2007). [ABSTRACT FROM AUTHOR]

Let G be a simple undirected connected graph. The Harary matrix of graph G , which is also called as the Reciprocal Distance matrix, was introduced by Plavsić et al. in 1993. Bapat and Panda, in 2018, defined the Reciprocal Distance Laplacian matrix, and Alhevaz et al., in 2019, defined the Reciprocal Distance signless Laplacian matrix. In this article, we find upper bounds and lower bounds for the spectral radius of the Reciprocal Distance matrix, Reciprocal Distance Laplacian matrix and Reciprocal Distance signless Laplacian matrix. [ABSTRACT FROM AUTHOR]

Let S (m, d, k) be the set of k-uniform supertrees with m edges and diameter d, and S1 (m, d, k) be the k-uniform supertree obtained from a loose path u1, e1, u2, e2, ..., ud, ed, ud+1 with length d by attaching m–d edges at vertex u⌊d/2⌋+1. In this paper, we mainly determine S1 (m, d, k) with the largest signless Laplacian spectral radius in S (m, d, k) for 3 ⩽ d ⩽ m − 1. We also determine the supertree with the second largest signless Laplacian spectral radius in S (m, 3, k). Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices). [ABSTRACT FROM AUTHOR]

In this paper, we obtain some bounds for the ABC spectral radius of general graphs. In continuing, we get some properties of the ABC eigenvalues of graphs, and then we establish some new bounds for the ABC energy E A B C. Finally, we set up the correlation between the ABC energy and other classes of graph energies. [ABSTRACT FROM AUTHOR]

The first and the second Zagreb index of a graph, usually defined as the sum of the squares of degrees over all vertices and the sum of the products of degrees of edge endvertices over all edges, respectively, are tightly related to the numbers of walks of length two and three in the graph. We provide here a lower bound on the spectral radius of adjacency matrix A of graph in terms of its Zagreb indices, based on the properties of the least square approximation of the vector A 2 j with the vectors Aj and j , where j is the all-one vector. The bound is sharp for all graphs with two main eigenvalues, surpassing the range of sharpness of other bounds among connected graphs. [ABSTRACT FROM AUTHOR]

In this paper, we present some spectral sufficient conditions for a graph to be Hamilton-connected in terms of the spectral radius or signless Laplacian spectral radius of the graph. Our results improve some previous work. [ABSTRACT FROM AUTHOR]

In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain the second lower bound on Wiener index for tricyclic graphs. As applications, those graphs with the first four minimum distance (resp. distance signless Laplacian) spectral radius among tricyclic graphs are characterized. [ABSTRACT FROM AUTHOR]

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k ⋅ P 3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k ⋅ P 3 as a subgraph, respectively. [ABSTRACT FROM AUTHOR]

The resistance matrix R = R (G) of G is a matrix whose (i, j)th entry is equal to the resistance distance r G (v i , v j) . The resistance distance spectral radius, ∂ 1 R (G) , of G is the largest eigenvalues of R. In this paper, we obtain several edge-grafting transformations which are decreasing and/or increasing resistance distance spectral radius. As applications of these transformations, some lower and upper bounds on resistance distance spectral radius are determined and the corresponding extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected connected graph. In this paper, new upper bounds on the distance Laplacian spectral radius of G are obtained. Moreover, new lower and upper hounds for the distance signless Laplacian spectral radius of G are derived. Some of the above mentioned bounds are sharp and the graphs attaining the corresponding bound are characterized. Several illustrative examples are included. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement, respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasicomplement, respectively. [ABSTRACT FROM AUTHOR]

In this paper, some fundamental properties of the Hadamard product of tensors are discussed. The closure of several classes of structured tensors under the Hadamard product is studied. As its important application, we present an upper bound on the spectral radius of the Hadamard product of nonnegative tensors. Moreover, we also provide some important inequalities on the spectral radius of the Hadamard products of the Hadamard powers for nonnegative tensors. Some numerical examples are reported to verify the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

We show that the spectral radius of a graph is an upper bound on the spectral norm of any of its orientations. We also show that equality is attained only when the orientation is sink-source. As an application, we find the extremal values of the spectral norm over the set of oriented trees, oriented unicyclic graphs and connected digraphs with n vertices and n arcs. [ABSTRACT FROM AUTHOR]

Abstract The spectral radius of a graph is the largest eigenvalue of the adjacency matrix corresponding to this graph. Let G be a graph with long pendant paths. In this note, we estimate the spectral radius of G , by obtaining appropriate upper and lower bounds. As an application, the spectral radius of some k-ary trees are estimated. [ABSTRACT FROM AUTHOR]

Abstract In a network or a graph, the total communicability (TC) has been defined as the sum of the entries in the exponential of the adjacency matrix of the network. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. In this paper, we obtain some bounds for total communicability of a graph G , T C (G) , in terms of spectral radius of the adjacency matrix, number of vertices, number of edges, minimum degree and the maximum degree of G. Moreover, we find some upper bounds for T C (G) when G is the Cartesian product, tensor product or the strong product of two graphs. In addition, Nordhaus–Gaddum-type results for the total communicability of a graph G are established. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a weighted digraph with vertex set V (G) = { v 1 , v 2 , ... , v n } and arc set E (G) , where the arc weights are nonzero nonnegative symmetric matrices. In this paper, we obtain an upper bound on the signless Laplacian spectral radius of a weighted digraph G , and if G is strongly connected, we also characterize the digraphs achieving the upper bound. Moreover, we show that an upper bound of weighted digraphs or unweighted digraphs can be deduced from our upper bound. [ABSTRACT FROM AUTHOR]

Let H be a k-uniform hypergraph on n vertices with degree sequence Δ = d 1 ≥ ⋯ ≥ d n = δ . E i denotes the set of edges of H containing i. The average 2-degree of vertex i of H is m i = ∑ { i , i 2 , ... i k } ∈ E i d i 2 ... d i k / d i k - 1 . In this paper, in terms of m i and d i , we give some upper bounds and lower bounds for the spectral radius of the signless Laplacian tensor ( Q (H) ) of H . Some examples are given to show the tightness of these bounds. [ABSTRACT FROM AUTHOR]

This paper gives lower and upper bounds on the covering radius of codes over ℤ p 2 with respect to Lee distance. We also determine the covering radius of various repetition codes over ℤ p 2 . [ABSTRACT FROM AUTHOR]

Using estimates in Sobolev spaces we prove that the solution to the Cauchy problem for the Camassa–Holm equation on the line with analytic initial data u 0 ( x ) and satisfying the McKean condition, that is the quantity m 0 ( x ) = ( 1 − ∂ x 2 ) u 0 ( x ) does not change sign, is analytic in the spatial variable for all time. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r ( t ) given by r ( t ) ≥ A − 1 ( 1 + C 1 B t ) − 1 exp { − C 0 ‖ u 0 ‖ H 1 t } , where A , B , C 1 and C 0 are suitable positive constants. [ABSTRACT FROM AUTHOR]

In this article, we establish some upper bounds for numerical radius inequalities, including those of 2 × 2 operator matrices and their off- diagonal parts. Among other inequalities, it is shown that if T = [Y/0 X/0], ... then and ... where X; Y are bounded linear operators on a Hilbert space H, r ≥ 1, and f, g are nonnegative continuous functions on [0,∞) satisfying the relation f(t)g(t) = t (t ∊ [0,∞)). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators T1, ... , Tn. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds on Z t -spectral radius of a tensor A ( t = 1 , 2 ) , which extend the upper bounds of Brauer to tensors. Moreover, an upper bound on the Z 1 -spectral radius is proposed via modulus sum of the entries of certain dimension of A , which improves the upper bound given by Li et al. Numerical experiments are given to illustrate the utility of the upper bound. [ABSTRACT FROM AUTHOR]

We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of disks is connected, and the sum of radii is minimized. We prove that the problem is NP-hard in planar weighted graphs if there are upper bounds on the radii and sketch a similar proof for planar point sets. For the case when there are no upper bounds on the radii, the complexity is open; we give a polynomial-time approximation scheme. We also give constant-factor approximation guarantees for solutions with a bounded number of disks; these results are supported by lower bounds, which are shown to be tight in some of the cases. Finally, we show that the problem is polynomially solvable if a connectivity tree is given, and we conclude with some experimental results. [ABSTRACT FROM AUTHOR]

Let $A_{1}, A_{2},\ldots, A_{k}$ be nonnegative matrices. In this paper, some upper bounds for the spectral radius $\rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k})$ are proposed. These bounds generalize some existing results, and comparisons between these bounds are also considered. [ABSTRACT FROM AUTHOR]

A relationship between s -extremal singly even self-dual [ 24 k + 8 , 12 k + 4 , 4 k + 2 ] codes and extremal doubly even self-dual [ 24 k + 8 , 12 k + 4 , 4 k + 4 ] codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, s -extremal singly even self-dual [ 56 , 28 , 10 ] codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length 24 k + 8 with covering radius meeting the Delsarte bound for k ≥ 137 . Similarly, we show that there is no extremal doubly even self-dual code of length 24 k + 16 with covering radius meeting the Delsarte bound for k ≥ 148 . We also determine the covering radii of some extremal doubly even self-dual codes of length 80. [ABSTRACT FROM AUTHOR]

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch. [ABSTRACT FROM AUTHOR]

Let $$-1 \le B

We present lower bounds for the uniform radius of spatial analyticity of solutions to the Korteweg-de Vries equation, which improve earlier results due to Bona, Grujić and Kalisch. [ABSTRACT FROM AUTHOR]

In this paper we give several expressions of spectral radius of a bounded operator on a Hilbert space, in terms of iterated λ -Aluthge transform, numerical radius and the asymptotic behavior of the powers of this operator. We also obtain several characterizations of normaloid operators. [ABSTRACT FROM AUTHOR]

Localized multiple kernel learning (LMKL) is an effective method of multiple kernel learning (MKL). It tries to learn the optimal kernel from a set of predefined basic kernels by directly using the maximum margin principle, which is embodied in support vector machine (SVM). However, LMKL does not consider the radius of minimum enclosing ball (MEB) which actually impacts the error bound of SVM as well as the separating margin. In the paper, we propose an improved version of LMKL, which is named ILMKL. The proposed method explicitly takes into consideration both the margin and the radius and so achieves better performance over its counterpart. Moreover, the proposed method can automatically tune the regularization parameter when learning the optimal kernel. Consequently, it avoids using the time-consuming cross-validation process to choose the parameter. Comprehensive experiments are conducted and the results well demonstrate the effectiveness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

For given graphs G and H , the graph G is H -saturated if G does not contain H as a subgraph but for any e ∈ E (G ‾) , G + e contains H. In this note, we prove that if G is an n -vertex K r + 1 -saturated graph such that for each vertex v ∈ V (G) , ∑ w ∈ N (v) d G (w) ≥ (r − 2) d (v) + (r − 1) (n − r + 1) , then ρ (G) ≥ ρ (S n , r) , where S n , r is the graph obtained from a copy of K r − 1 with vertex set S by adding n − r + 1 vertices, each of which has neighborhood S. This provides a sharp lower bound for the spectral radius in an n -vertex K r + 1 -saturated graph for r = 2 , 3 , verifying a special case of a conjecture by Kim, Kim, Kostochka and O. [ABSTRACT FROM AUTHOR]

We obtain the sharp bounds for the spectral radius of a nonnegative matrix and then obtain some known results or new results by applying these bounds to a graph or a digraph and revise and improve two known results. [ABSTRACT FROM AUTHOR]

The inverse nodal problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from zeros for eigenfunctions corresponding to the transmission eigenvalues. It is shown that the space of potential functions q which correspond to interior transmission problems characterized by Ω : = { q : q ∈ L 2 [ 0 , 1 ] } , under a certain metric, is homeomorphic to the partition set of the space of quasinodal sequences. As a consequence, the inverse nodal problem defined on the partition set of admissible sequence, is stable. [ABSTRACT FROM AUTHOR]

We consider the critical heat equation in where is a smooth bounded domain in and is a ball of of center and radius small. We show that if is small enough, then there exists a sign-changing stationary solution of (CH) such that the solution of (CH) with initial value blows up if is sufficiently small. This shows, in particular, that the set of the initial conditions for which the solution of (CH) is global and bounded is not star-shaped. [ABSTRACT FROM AUTHOR]

We apply certain matrix inequalities involving eigenvalues, the numerical radius, and the spectral radius to obtain new bounds and majorization relations for the zeros of a class of Fibonacci-type polynomials. Our results improve upon some earlier bounds for the zeros of these polynomials. [ABSTRACT FROM AUTHOR]

In this study, a new adaptive trust-region strategy is presented to solve nonlinear systems. More specifically, we propose a new method leading to produce a smaller trust-region radius close to the optimizer and a larger trust-region radius far away from the optimizer. Accordingly, it can lead to a smaller step-size close to the optimizer and a larger one far away from the optimizer. The new strategy includes a convex combination of the maximum norm of function value of some preceding successful iterates and the current norm of function value. The global convergence of the proposed approach is established while the local q-quadratic convergence rate is proved under local error bound condition, which is weaker than the nonsingularity. Numerical results of the proposed algorithm are also reported. [ABSTRACT FROM AUTHOR]

A k-radius sequence over an n-element alphabet A is a sequence in which every two elements of A appear within distance at most k (where the distance is defined as the difference of indices). By a k-radius sequence over an n-element alphabet A we mean a sequence in which every two elements of A appear within distance at most k. The problem of constructing shortest possible k-radius sequences, motivated by some problems occurring in large data transfer, has been studied by several authors recently. In this paper we present an explicit construction of "short" k-radius sequences for some values of k and n. This construction allows us to find 2-radius sequences of the shortest possible length for all but very special values of n. For all n we construct 2-radius sequences whose length differs from the length of the shortest one only by a constant. Moreover, we construct shortest possible k-radius sequences when n = 2k² + 2k + 1 and k is a power of a prime. Our construction depends on the existence of some other sequences that we call k-perfect and k-additive. We investigate these sequences as they seem to be interesting in themselves. [ABSTRACT FROM AUTHOR]

Using upper bounds for the signless Laplacian spectral radius of graphs established by Yu et al. [9], Wang et al. [8], Liu et al. [7], and Li et al. [5], we in this note present sufficient conditions which are based on the signless Laplacian spectral radius for some Hamiltonian properties of graphs. [ABSTRACT FROM AUTHOR]

Edge betweenness is a measure of the influence of edges in networks. The edge betweenness of an edge is defined to be the number of shortest paths between pairs of vertices that run along it. For a given graph $$G=(V(G),E(G))$$ , the adjusted betweenness centrality $$c_v^*$$ of a vertex $$v\in V(G)$$ is defined to be the sum of the edge betweenness of all the edges incident to $$v$$ . In this paper we study the mathematical properties of $$c_v^*$$ for vertices in trees. We present lower and upper bounds for $$c_v^*$$ in terms of the number of vertices, vertex degree, eccentricity, transmission, maximum degree, diameter, and radius. [ABSTRACT FROM AUTHOR]

Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Z -eigenpair of a tensor, in particular, an irreducible nonnegative tensor. We present some bounds for the eigenvector and Z -spectral radius. The proposed bounds improve some existing ones. An example with practical applications is given to show the proposed bound. [ABSTRACT FROM AUTHOR]