Search

A fractional matching of a graph G is a function f : E (G) → [ 0 , 1 ] such that for any v ∈ V (G) , ∑ e ∈ E G (v) f (e) ≤ 1 , where E G (v) = { e ∈ E (G) : e is incident with v in G }. The fractional matching number of G is μ f (G) = max { ∑ e ∈ E (G) f (e) : f is a fractional matching of G }. Let k ∈ (0 , n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee μ f (G) > n − k 2 in a graph with minimum degree δ , which implies the result on the fractional perfect matching due to Fan et al. (2022) [6]. For a set { A , B , C , ... } of graphs, an { A , B , C , ... } -factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of { A , B , C , ... }. We present a tight sufficient condition in terms of the spectral radius for the existence of a { K 2 , { C k } } -factor in a graph with minimum degree δ , where k ≥ 3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a { K 1 , 1 , K 1 , 2 , ... , K 1 , k } -factor with k ≥ 2 in a graph with minimum degree δ , which generalizes the result of Miao et al. (2023) [10]. [ABSTRACT FROM AUTHOR]

A graph G is called fractional [ a , b ] -covered if for every edge e of G there is a fractional [ a , b ] -factor with the indicator function h such that h (e) = 1. In this paper, we provide a tight spectral radius condition for graphs being fractional [ a , b ] -covered. [ABSTRACT FROM AUTHOR]

In this article we present new numerical radius inequalities for sectorial matrices, and their analogues. For example, we show that if A is a sectorial matrix with sectorial index γ ∈ [ 0 , π 2) , then ‖ A A ⁎ + A ⁎ A ‖ 2 (1 + sin 2 ⁡ γ) ≤ w (A) 2 , where w (A) is the numerical radius of A. The significance of this result is the geometric meaning of the obtained inequality. In particular, we will show how this interpolates some well known inequalities in the literature, in a way that depends on the angle of the sector that contains the numerical range of the sectorial matrix A. Many other interpolating inequalities will be presented to show how the ratio ‖ A ‖ w (A) depends mainly on the angle of the sector that contains the numerical range of the matrix A. Our discussion will lead to a new vision about those matrices which contain the origin in the interior of their numerical ranges. [ABSTRACT FROM AUTHOR]

We show that the kite graph K 4 (n) uniquely maximizes the distance spectral radius among all connected 4-chromatic planar graphs on n vertices. [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]

The ABC matrix of a graph G , proposed by Estrada in 2017, can be regarded as a weighed version of adjacency matrices of graphs, in which the (u , v) -entry is equal to d u + d v − 2 d u d v if uv is an edge of a graph G , and 0 otherwise, where d u represents the degree of u in G. The research about ABC spectral radius (largest eigenvalue of ABC matrix) of graphs is rather active in recent years. In this paper, we characterize the bicyclic graphs with the first two maximum ABC spectral radii, which are just the unique two bicyclic graphs of order n with maximum degree n − 1 if n ≥ 7. [ABSTRACT FROM AUTHOR]

A perfect matching in a graph G is a set of disjoint edges covering all vertices of G. Let ρ (G) be the spectral radius of a graph G , and let θ (n) be the largest root of x 3 − (n − 4) x 2 − (n − 1) x + 2 (n − 4) = 0. In this paper, we prove that for a positive even integer n ≥ 8 or n = 4 , if G is an n -vertex graph with ρ (G) > θ (n) , then G has a perfect matching; for n = 6 , if ρ (G) > 1 + 33 2 , then G has a perfect matching. It is sharp for every positive even integer n ≥ 4 in the sense that there are graphs H with ρ (H) = θ ′ (n) and no perfect matching, where θ ′ (n) = θ (n) if n = 4 or n ≥ 8 and θ ′ (6) = 1 + 33 2. [ABSTRACT FROM AUTHOR]

In this paper, we investigate how the adjacency spectral radius and signless Laplacian spectral radius behave when a connected uniform hypergraph is perturbed by grafting edges. We extend the classical theorem of Li and Feng (1979) [10] about spectral radius from connected graphs to connected uniform hypergraphs by using a constructive method. This result also generalizes the results of Cvetković and Simić (2009) [2] , and Su et al. (2018) [22]. As applications, we determine the k -uniform supertrees of order n with the first two smallest adjacency spectral radii (signless Laplacian spectral radii, respectively). Also, we determine the k -uniform supertrees of order n with the first two smallest Laplacian spectral radii, in the case when k is even. [ABSTRACT FROM AUTHOR]

The rank and the spectral radius of a graph is the rank and the spectral radius of its adjacency matrix, respectively. In this paper we find the graphs with extremal spectral radius over the set of all connected graphs with n vertices and rank 4. [ABSTRACT FROM AUTHOR]

In this paper, we present two new sufficient conditions on the spectral radius ρ (G) that guarantee the hamiltonicity and traceability of a k -connected graph G of sufficiently large order, respectively, unless G is a specified exceptional graph. In particular, if k ≥ 2 , n ≥ k 3 + k + 2 , and ρ (G) > n − k − 1 − 1 n , then G is hamiltonian, unless G is a specified exceptional graph. If k ≥ 1 , n ≥ k 3 + k 2 + k + 3 , and ρ (G) > n − k − 2 − 1 n , then G is traceable, unless G is a specified exceptional graph. [ABSTRACT FROM AUTHOR]

In [1] , J.Y. Shao proposed the general product of tensors. Based on this product, the Minc-type bound on spectral radius of nonnegative tensors is given. And the generalized Minimax Theorem of nonnegative tensors is presented. Moreover, we give some bounds on the spectral radius of the general product of tensors. [ABSTRACT FROM AUTHOR]

For a graph G , the spectral radius ρ (G) of G is the largest eigenvalue of its adjacency matrix. The coalescence of two graphs H with a root v and K with a root w is obtained by identifying v and w from disjoint union of H and K. In this paper, we investigate the upper bounds of the spectral radius of the coalescence of two graphs, which generalize some results by Passbani and Salemi in 2019. Furthermore, we show that if the graph G u v ⁎ is obtained from a connected graph G by contracting the internal edge uv which not contained in any triangle of G , then ρ (G) ≤ ρ (G u v ⁎) , the corresponding extremal graphs are characterized completely, and also extend a result of Hoffman and Smith. As an application, a new sharp upper bound on the spectral radius of a tree is provided. [ABSTRACT FROM AUTHOR]

Let G be a simple graph. We say that a hypergraph H is a Berge- G if there is a bijection ϕ : E (G) → E (H) such that e ⊂ ϕ (e) for all e ∈ E (G). For any r -uniform hypergraph H and a real number p ≥ 1 , the p -spectral radius λ (p) (H) of H is defined as λ (p) (H) : = max ‖ x ‖ p = 1 ⁡ r ∑ { i 1 , i 2 , ... , i r } ∈ E (H) x i 1 x i 2 ⋯ x i r . In this paper we study the p -spectral radius of Berge- G (G ∈ C n +) hypergraphs and determine the 3-uniform hypergraphs with maximum p -spectral radius for p ≥ 1 among all Berge- G (G ∈ C n +) hypergraphs, where C n + is the set of graphs of order n obtained from C n by adding an edge. [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]

Let H n (r) be the set of all connected r -graphs with given size n. In this paper, we investigate the effect on the spectral radius of r -uniform hypergraphs by grafting or contracting an edge and then give the ordering of the r -graphs with small spectral radius over H n (r) , when n ≥ 20. [ABSTRACT FROM AUTHOR]

The ABC matrix of a graph G is a square matrix, where the (i , j) -entry A B C i j = d i + d j − 2 d i d j if there is an edge between vertices i and j in G , and A B C i j = 0 otherwise. The ABC spectral radius of G , denoted by ν 1 (G) , is the largest eigenvalue of the ABC matrix of G. In this paper, we prove that for a unicyclic graph G of order n ≥ 4 , 2 = ν 1 (C n) ≤ ν 1 (G) ≤ ν 1 (S n + e) , with equality if and only if G ≅ C n for the lower bound, and if and only if G ≅ S n + e for the upper bound. This solves a conjecture by Ghorbani et al. [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]

The spectral radius of a uniform hypergraph G is the maximum modulus of the eigenvalues of the adjacency tensor of G. For k ≥ 2 , among connected k -uniform hypergraphs with m ≥ 1 edges, we show that the k -uniform loose path with m edges is the unique one with smallest spectral radius, and we also determine the unique ones with second smallest spectral radius when m ≥ 2. [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]

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]

A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H -free graph of order n ? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k = 2. [ABSTRACT FROM AUTHOR]

In this paper we show how the α -spectral radius changes under the edge grafting operations on connected k -uniform hypergraphs. We characterize the extremal hypertree for α -spectral radius among k -uniform non-caterpillar hypergraphs with given order, size and diameter. We also characterize the supertree with the second largest α -spectral radius among all k -uniform supertrees on n vertices. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n and also let ρ (G) be the spectral radius of the graph G. In this paper, we prove that ρ (G) ≤ ρ 2 (G − v k) + 2 d k − 1 for any non-isolated vertex v k of degree d k in G , thus confirming a conjecture posed by Guo et al. (2019) [3]. Moreover, we characterized all connected graphs for which this bound is attained. [ABSTRACT FROM AUTHOR]

Let T ∈ B (H) be a bounded linear operator and ω (T) denotes the numerical radius of T. Let S 1 , ... , S m , T 1 , ... , T m ∈ B (H) such that S i T j = T j S i and S i ⁎ T j = T j S i ⁎ for all 1 ≤ i , j , ≤ m. It is shown that ω (∑ i = 1 m S i T i) ≤ 1 2 ‖ ∑ i = 1 m S i ⁎ S i + S i S i ⁎ ‖ 1 2 ‖ ∑ i = 1 m T i ⁎ T i + T i T i ⁎ ‖ 1 2 . Moreover, we show that ω (∑ i = 1 m S i T i + T i S i) ≤ ‖ ∑ i = 1 m S i ⁎ S i + S i S i ⁎ ‖ 1 2 ‖ ∑ i = 1 m T i ⁎ T i + T i T i ⁎ ‖ 1 2 , for all S 1 , ... , S m , T 1 , ... , T m ∈ B (H). In particular, for m = 1 , we refine some well known inequalities. [ABSTRACT FROM AUTHOR]

In this article, we present a special treatment of sectorial matrices under Lieb functions. This treatment results from a new characterization of sectorial matrices using blocks, which then implies a new set of inequalities governing such matrices, where singular values, determinants, numerical radius and unitarily invariant norms inequalities are retrieved in a general form that implies many well known results in the literature. [ABSTRACT FROM AUTHOR]

For a connected graph G and α ∈ [ 0 , 1 ] , let D α (G) be the matrix D α (G) = α Tr (G) + (1 − α) D (G) , where D (G) is the distance matrix of G and Tr (G) is the diagonal matrix of its vertex transmissions. Let K m be a complete graph of order m. For n , s fixed, n > s , let G p = K s ∨ (K p ∪ K n − s − p) be the graph obtained from K s and K p ∪ K n − s − p and the edges connecting each vertex of K s with every vertex of K p ∪ K n − s − p. This paper presents some extremal results on the spectral radius of D α (G) that generalize previous results on the spectral radii of the distance matrix and distance signless Laplacian matrix. Among all connected graphs G on n vertices with a vertex/edge connectivity at most s , it is proved that 1. there exists a unique α _ ∈ (3 4 , 3 n − s 4 n − 2 s) such that if α ∈ [ 0 , α _) then the minimal spectral radius of D α (G) is uniquely attained by G = G 1 , 2. there exists a unique α ‾ ∈ (3 4 , 3 n − s 4 n − 2 s) , α ‾ ≥ α _ , such that if α ∈ (α ‾ , 1) then the minimal spectral radius of D α (G) is uniquely attained by G = G ⌊ n − s 2 ⌋ , and 3. if α = 1 then the minimal spectral radius of Tr (G) is n − 1 + ⌈ n − s 2 ⌉ and it is uniquely attained by G = G ⌊ n − s 2 ⌋. Furthermore, in terms of n and s , a tight lower bound l (n , s) of α _ and a tight upper bound u (n , s) of α ‾ are obtained. Finally, for s fixed, it is observed that lim n → ∞ ⁡ l (n , s) = lim n → ∞ ⁡ α _ = lim n → ∞ ⁡ u (n , s) = lim n → ∞ ⁡ α ‾ = 3 4. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with vertices v 1 , ... , v n. The distance matrix of G , denoted by D (G) , is the n × n matrix whose (i , j) -entry is equal to d (v i , v j) (length of a shortest path between v i and v j). By the distance spectral radius of G that is denoted by μ (G) , we mean the largest eigenvalue of the distance matrix of G. Let X = (m 1 , ... , m t) and Y = (n 1 , ... , n t) , where m 1 ≥ ⋯ ≥ m t ≥ 1 and n 1 ≥ ⋯ ≥ n t ≥ 1 are integer. We say X majorizes Y and let X ⪰ M Y , if for every j , 1 ≤ j ≤ t − 1 , ∑ i = 1 j m i ≥ ∑ i = 1 j n i with equality if j = t. In this paper we find a relation between the majorization and the distance spectral radius of complete multipartite graphs. We show that if (m 1 , ... , m t) ⪰ M (n 1 , ... , n t) and (m 1 , ... , m t) ≠ (n 1 , ... , n t) then μ (K m 1 , ... , m t ) > μ (K n 1 , ... , n t ) , where K n 1 , ... , n t is the complete multipartite graph with t parts of size n 1 , ... , n t. Using the above inequality we find that among all complete multipartite graphs with n vertices and t ≥ 2 parts, the Turán graphs have the minimum distance spectral radius and the split graphs have the maximum distance spectral radius. Let t ≥ 2 and n 1 , ... , n t be some positive integers. We show that n + a + b − 4 + (n + a + b) 2 − 4 a b (t + 1) 2 ≤ μ (K n 1 , ... , n t ) ≤ 2 n − t − 2 + (2 n − 2 t + 1) 2 + t 2 − 1 2 , where n = n 1 + ⋯ + n t , a = ⌈ n t ⌉ and b = ⌊ n t ⌋. In addition we investigate the equality in both sides. Finally we obtain that μ (K n 1 , ... , n t ) ≤ 2 n − t − 1. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple connected graph of order n and size m , and let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 > ρ n = 0 , be the normalized Laplacian eigenvalues of G. The greatest eigenvalue, ρ 1 , is referred to as the normalized Laplacian spectral radius. The Laplacian incidence energy, the Randić energy, and Kemeny's constant of G , are defined by L I E (G) = ∑ i = 1 n − 1 ρ i , R E (G) = ∑ i = 1 n | ρ i − 1 | , and K (G) = ∑ i = 1 n − 1 1 ρ i , respectively. New lower bounds for the normalized Laplacian spectral radius are obtained. The acquired bounds are then used to obtain new bounds for the L I E (G). In addition, some relations between L I E (G) , K (G) , R E (G) are determined. [ABSTRACT FROM AUTHOR]

Let G ‾ n , k denote the set of strongly connected digraphs with order n and arc connectivity k , and let G ‾ n , k ⁎ denote the set of digraphs in G ‾ n , k with all vertices having outdegree and indegree greater than k. In this paper, we determine the unique digraph with the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k. We also determine the unique one with the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ with k = 1 , 2. For the general case, we propose a conjecture on the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎. Moreover, we characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k. We also characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ with k = 1 , 2. For the general case, we propose a conjecture on the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎. [ABSTRACT FROM AUTHOR]

We prove several Hilbert-Schmidt numerical radius inequalities for operator matrices. A refinement of the triangle inequality for the Hilbert-Schmidt norm is also given. [ABSTRACT FROM AUTHOR]

Let A be a positive bounded operator on a Hilbert space (H , 〈 ⋅ , ⋅ 〉). The semi-inner product 〈 x , y 〉 A : = 〈 A x , y 〉 , x , y ∈ H induces a semi-norm ‖ ⋅ ‖ A on H. Let ‖ T ‖ A and w A (T) denote the A -operator semi-norm and the A -numerical radius of an operator T in semi-Hilbertian space (H , ‖ ⋅ ‖ A) , respectively. In this paper, we prove the following characterization of w A (T) w A (T) = sup α 2 + β 2 = 1 ⁡ ‖ α T + T ♯ A 2 + β T − T ♯ A 2 i ‖ A , where T ♯ A is a distinguished A -adjoint operator of T. We then apply it to find upper and lower bounds for w A (T). In particular, we show that 1 2 ‖ T ‖ A ≤ max ⁡ { 1 − | cos ⁡ | A 2 T , 2 2 } w A (T) ≤ w A (T) , where | cos ⁡ | A T denotes the A -cosine of angle of T. Some upper bounds for the A -numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given. [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 We define a norm on the space of bounded linear operators on a Hilbert space, which generalizes the numerical radius norm. We investigate basic properties of this norm and prove inequalities involving it. A concrete example of this norm is also given. [ABSTRACT FROM AUTHOR]

Abstract Let λ (D) be the A α spectral radius of digraph D , and let G n r be the set of digraphs with order n and dichromatic number r. In this paper, we characterize the digraph which has the maximal A α spectral radius in G n r. Moreover, we also determine the unique digraph which attains the maximum (resp. minimum) A α spectral radius among all strongly connected bicyclic digraphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov (2017) [7] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let u and v be two vertices of a connected graph G . Suppose that u and v are connected by a path w 0 ( = v ) w 1 ⋯ w s − 1 w s ( = u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1 . Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Let s = 0 , 1 . Nikiforov and Rojo (2018) [9] conjectured that ρ α ( G p , s , q ( u , v ) ) < ρ α ( G p − 1 , s , q + 1 ( u , v ) ) if p ≥ q + 2 . In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A α -spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A α -spectral radius with fixed order and matching number. These results generalize some known results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of an m -order n -dimensional generalized Hilbert tensor H n = ( H i 1 i 2 ⋯ i m ) , H i 1 i 2 ⋯ i m = 1 i 1 + i 2 + ⋯ i m − m + a , a ∈ R ∖ Z − ; i 1 , i 2 , ⋯ , i m = 1 , 2 , ⋯ , n , and show that its H -spectral radius and its Z -spectral radius are smaller than or equal to M ( a ) n m − 1 and M ( a ) n m 2 , respectively, here M ( a ) is a constant depending on a . Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for a ≥ 1 . For an m -order infinite dimensional generalized Hilbert tensor H ∞ with a > 0 , we prove that H ∞ defines a bounded and positively ( m − 1 ) -homogeneous operator from l 1 into l p ( 1 < p < ∞ ) . Two upper bounds on the norms of corresponding positively homogeneous operators are obtained. [ABSTRACT FROM AUTHOR]

Let t ≥ 3 and G be a graph of order n , with no K 2 , t minor. If n > 400 t 6 , then the spectral radius μ ( G ) satisfies μ ( G ) ≤ t − 1 2 + n + t 2 − 2 t − 3 4 , with equality if and only if n ≡ 1 ( mod t ) and G = K 1 ∨ ⌊ n / t ⌋ K t . For t = 3 the maximum μ ( G ) is found exactly for any n > 40000 . [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]

V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on the entries of B . We calculate the radius of the neighborhood U . A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field Q p of p -adic numbers and the field F ( ( T ) ) of Laurent series over a field F . [ABSTRACT FROM AUTHOR]

We study the effect of three types of graft transformations to increase or decrease the distance spectral radius of connected uniform hypergraphs, and we determine the unique k -uniform hypertrees with maximum, second maximum, minimum and second minimum distance spectral radius, respectively. [ABSTRACT FROM AUTHOR]

It is known that the numerical radius of the Hadamard product A ∘ B of two n -by- n matrices A and B is related to those of A and B by (a) w ( A ∘ B ) ≤ 2 w ( A ) w ( B ) , (b) w ( A ∘ B ) ≤ w ( A ) w ( B ) if one of A and B is normal, and (c) w ( A ∘ B ) ≤ ( max i ⁡ a i i ) w ( B ) if A = [ a i j ] i , j = 1 n is positive semidefinite. In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining A = [ 0 0 a 0 ] and B = [ 0 0 b 0 ] for (a), A = [ a 1 0 0 a 2 ] ( | a 1 | ≥ | a 2 | ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (b), and A = [ a 1 a 3 a 2 a 4 ] ≥ 0 ( a 1 ≥ a 4 ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (c). [ABSTRACT FROM AUTHOR]

For two n -by- n matrices A and B , it was known before that their numerical radii satisfy the inequality w ( A B ) ≤ 4 w ( A ) w ( B ) , and the equality is attained by the 2-by-2 matrices A = [ 0 1 0 0 ] and B = [ 0 0 1 0 ] . Moreover, the constant “4” here can be reduced to “2” if A and B commute, and the corresponding equality is attained by A = I 2 ⊗ [ 0 1 0 0 ] and B = [ 0 1 0 0 ] ⊗ I 2 . In this paper, we give a complete characterization of A and B for which the equality holds in each case. More precisely, it is shown that w ( A B ) = 4 w ( A ) w ( B ) (resp., w ( A B ) = 2 w ( A ) w ( B ) for commuting A and B ) if and only if either A or B is the zero matrix, or A and B are simultaneously unitarily similar to matrices of the form [ 0 a 0 0 ] ⊕ A ′ and [ 0 0 b 0 ] ⊕ B ′ (resp., ⊕ A ′ and ⊕ B ′ ) with w ( A ′ ) ≤ | a | / 2 and w ( B ′ ) ≤ | b | / 2 . An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B , we show that w ( A B ) = w ( A ) w ( B ) if and only if either A or B is a scalar matrix, or A and B are simultaneously unitarily similar to [ a 1 0 0 a 2 ] and [ b 1 0 0 b 2 ] with | a 1 | ≥ | a 2 | and | b 1 | ≥ | b 2 | . [ABSTRACT FROM AUTHOR]

A connected graph G with maximum degree Δ and edge chromatic number χ ′ ( G ) = Δ + 1 is called Δ-critical if χ ′ ( G − e ) = Δ for every edge e of G . In this paper, we consider two weaker versions of Vizing's conjecture, which concern the spectral radius ρ ( G ) and the signless Laplacian spectral radius μ ( G ) of G . We obtain some lower bounds for ρ ( G ) and μ ( G ) , and present some cases where the conjectures are true. Finally, several open problems are also proposed. [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]

We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

We determine the unique graphs with minimum distance Laplacian spectral radius among connected graphs with fixed number of pendent vertices, the unique trees with minimum distance Laplacian spectral radius among trees with fixed bipartition, the unique graphs with minimum distance Laplacian spectral radius among graphs with fixed edge connectivity at most half of the number of vertices. We also discuss the minimum distance Laplacian spectral radius of graphs with fixed connectivity. For k = 1 , … , ⌊ n − 2 2 ⌋ , we determine the unique n -vertex tree with the ( k + 1 ) -th smallest distance Laplacian spectral radius. [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 suggest a novel method to determine the joint spectral radius of finite sets of matrices by validating the finiteness property. It is based on finding a certain finite tree with nodes representing sets of matrix products. Our approach accounts for cases where one or several matrix products satisfy the finiteness property. Moreover, is potentially functional even for reducible sets of matrices. [ABSTRACT FROM AUTHOR]