Allem, L. Emilioda Silveira, Lucas G. M.Trevisan, Vilmar
24页
查看更多>>摘要：In 1973 Schwenk [7] proved that almost every tree has a cospectral mate. Inspired by Schwenk's result, in this paper we study the spectrum of two families of trees. The p-sun of order 2p + 1 is a star K-1,K-p, with an edge attached to each pendant vertex, which we show to be determined by its spectrum among connected graphs. The (p, q)-double sun of order 2(p + q + 1) is the union of a p-sun and a q-sun by adding an edge between their central vertices. We determine when the (p, q)-double sun has a cospectral mate and when it is determined by its spectrum among connected graphs. Our method is based on the fact that these trees have few distinct eigenvalues and we are able to take advantage of their nullity to shorten the list of candidates. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：This work presents a parametrized family of distances, namely the Alpha Procrustes distances, on the set of symmetric, positive definite (SPD) matrices. The Alpha Procrustes distances provide a unified formulation encompassing both the Bures-Wasserstein and Log-Euclidean distances between SPD matrices. We show that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of SPD matrices, which encompass both the Log-Euclidean and Wasserstein Riemannian metrics. This formulation is then generalized to the set of positive definite Hilbert-Schmidt operators on a Hilbert space, unifying the infinite-dimensional BuresWasserstein and Log-Hilbert-Schmidt distances. In the setting of reproducing kernel Hilbert spaces (RKHS) covariance operators, we obtain closed form formulas for all the distances via the corresponding kernel Gram matrices. From a statistical viewpoint, the Alpha Procrustes distances give rise to a parametrized family of distances between Gaussian measures on Euclidean space, in the finite-dimensional case, and separable Hilbert spaces, in the infinite-dimensional case, encompassing the 2-Wasserstein distance, with closed form formulas via Gram matrices in the RKHS setting. The presented formulations are new both in the finite and infinite dimensional settings. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：A B S T R A C T By using a variational principle we find a necessary and sufficient condition for an operator to majorise the parallel sum of two positive definite operators. This result is then used as a vehicle to create new operator inequalities involving the parallel sum.(c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：A matrix mean sigma is said to satisfy the in-betweenness property with respect to the metric d if for any pair of positive definite matrices A and B,& nbsp;d(A, A sigma B) <= d(A, B).& nbsp;In this article, we introduce the geodesic in-betweenness property for k-tuples of positive definite matrices with respect to any metric. Moreover, we show that in spaces of non-positive curvature, geodesic in-betweenness implies in-betweenness in the case of two matrices. We then show some examples of metrics and means for which this property is satisfied. (C)& nbsp;2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties of RiordanKrylov matrices that are analogous to those of Riordan matrices are developed. These results and techniques are used to study incidence algebras of a poset as well as the chain and zeta-multichain polynomials of the poset. Throughout the paper applications to enumeration problems in combinatorics and number theory are provided.(c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：A (k, g)-cage is a k-regular simple graph of girth g with minimum possible number of vertices. In this paper, (k, g)-cages which are Moore graphs are referred as minimal (k, g)-cages. A simple connected graph is called distance regular (DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular (DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal (k, g)-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal (k, g)-cages we give a formula for distance spectral radius in terms of k and g, and also determine polynomials of degree [g/2] , which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal (k, g)-cage of diameter d has d + 1 distinct distance eigenvalues, and this partially answers a problem posed in [1]. We prove that every DBR graph is a 2-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal (k, g)-cages. Finally we determine the full distance spectrum of subdivision of some minimal (k, g)-cages.(C)& nbsp;2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let G be a simple graph with vertex set V and edge set E, and let di be the degree of the vertex v(i) is an element of & nbsp;V. If the vertices vi and v(j) are adjacent, we denote the respective edge by v(i)v(j) & nbsp;is an element of & nbsp; E. A vertex-degree-based topological index phi is defined as phi(G) = sigma(vivj is an element of E)& nbsp;phi(di, dj), where phi(i,j )is a function with the & nbsp;(& nbsp;)property phi(i,j) = phi(j,i). The general extended adjacency matrix A(phi) is defined as [A(phi)](ij )= phi(di,dj) if v(i)v(j)& nbsp;is an element of & nbsp;E, and 0 otherwise. The energy associated to phi of G is the sum of the absolute values of the eigenvalues of A(phi).& nbsp;In this paper we show that rho(G) epsilon(phi)(G) >=& nbsp;2(phi)(G) for all connected graphs G, where rho(G) is the spectral radius of G and phi(a,b) &NOTEQUexpressionL; 0 for all a, b & nbsp;is an element of & nbsp; N. We also characterize the graphs where equality holds. As a consequence, for any tree T with n vertices, E-phi(T)& nbsp;>=& nbsp;& nbsp;root 2./n - 1 phi(1,(n-1)), with equality holding if and only if T expressionpproximexpressiontely equexpressionl to & nbsp;S-n. (C)& nbsp;2021 Elsevier Inc. All rights reserved.& nbsp;
Akbari, SaieedAlazemi, AbdullahAndelic, MilicaHosseinzadeh, Mohammad Ali...
11页
查看更多>>摘要：The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree delta(G) and minimum degree delta(G) whose adjacency matrix is non-singular, E (G) >=& nbsp;delta(G) + delta(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G)) >= |& nbsp;E(G)|& nbsp;& nbsp;+ delta(G) - 5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. (C)& nbsp;2021 Elsevier Inc. All rights reserved.
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