查看更多>>摘要：For a graph G, we associate a family of real symmetric matrices, S(G), where for any A is an element of S(G), the location of the nonzero off-diagonal entries of Aare governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we give a characterization of all connected threshold graphs Gwith q(G) = 2. Moreover, we study the values of q( G) for connected threshold graphs with trace 2, 3, n - 2, n - 3, where nis the order of threshold graph. The values of q(G) are determined for all connected threshold graphs with 7 and 8 vertices with two exceptions. Finally, a sharp upper bound for q(G) over all connected threshold graph Gis given. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The alpha-normal labelling method, which was introduced by Lu and Man in 2014, is an effective method for comparing the adjacency spectral radii of uniform hypergraphs. Recently, the aspectra of graphs and hypergraphs have attracted much attention from researchers. In this paper, we present a new kind of generalization of alpha-normal labelling method relating to the alpha-spectral radius of k-uniform hypergraphs. The effect of several types of graph operations on the aspectral radius of the hypergraph are studied. As applications, we determine the unique hypergraph with the minimum aspectral radius among connected k-uniform hypergraphs with given edge number, among c-cyclic hypergraphs with c >= 1, respectively, and determine the unique supertree with second minimum alpha-spectral radius. Furthermore, we extend a recent result obtained by Wang and Yuan on the minimum spectral radius of supertrees with some constraints on maximal degree. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let M-n(F) denote the ring of n x n matrices with entries from a field F. For a subset S subset of M-n(F), the null ideal N(S) of Sis the set of all polynomials fwith coefficients in M-n(F) such that f(s) = 0 for all s is an element of S. We investigate conditions on Sunder which N(S) is a two-sided ideal of the polynomial ring M-n(F)[x]. In particular, we describe all finite subsets S subset of M-2(F) for which N(S) is a two-sided ideal. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The (real) minimum semidefinite rank of a signed graph is the minimum rank among all real symmetric positive semidefinite matrices associated to the graph and having the given sign pattern. We give a new lower bound for the minimum semidefinite rank of a signed multigraph and show it equals a new upper bound for signed complete multigraphs. This allows a complete characterization of signed multigraphs with minimum semidefinite rank two. We also determine the minimum semidefinite rank of all signed wheel graphs. (C) 2022 Elsevier Inc. All rights reserved.
Fabila-Carrasco, John StewartLledo, FernandoPost, Olaf
15页
查看更多>>摘要：In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the underlying graph. In particular, we give a family of spectral obstructions parametrised by the magnetic potential for the graph to be matchable (i.e., having a perfect matching) or for the existence of a Hamiltonian cycle. We base our analysis on a special case of the spectral preorder introduced in [8], and we use the magnetic potential as a spectral control parameter. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let (a) = a(0), a(1), a(2),... be a sequence over any finite field Fwith a(0)= 0. For each positive integer n, let A(n) be the associated n x n skew-symmetric Toeplitz matrix [a(0) a(1) a(2) a(3) . . . a(n-1) - a(1) a(0) a(1) a(2) . . . a(n-2) -a(2) -a(1) a(0) a(1) . . . a(n-3) . . . . . . . . . . . . . . . -a(n-1) -a(n-2) -a(n-3)...... a(0)] If the sequence is eventually periodic but not mirrorperiodic, then the nullitysequence {nu(n)= null(A(n)) : n is an element of N} is also eventually periodic, where nu(n)= null(A(n)) is the nullity of the matrix A(n). For s a certain multiple of the period of the nullity sequence, a recursion formula produces the vectors in ker(A(n+qs)) from those in ker(A(n)), for nsufficiently large and for non-negative integers q. Published by Elsevier Inc.
查看更多>>摘要：Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix Ais an n xn pseudo-involution then the singular values of Amust come in reciprocal pairs. Moreover, we give a complete analysis of existence and nonexistence of the eigenvectors of Riordan matrices. This leads to a surprising partition of the group of Riordan matrices into matrices with three different types of sets of eigenvectors. Finally, given a nonzero vector v, we investigate the Riordan matrices Athat stabilize the vector v, i.e. Av = v. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The topic of this paper lies between algebraic theory of *-rings and *-algebras on one side, and analytic theory of C*-algebras on the other side. A map theta : A -> B between unital *-rings is called range orthogonal isomorphism if it is bijective and preserves range orthogonality in both directions. We show that any additive (resp. linear) range orthogonal isomorphism is canonical, that is, it is a *-isomorphism followed by multiplication from the right by an invertible element, provided that Ais generated by projections as a *-ring. In case of general * rings and *-algebras we show that direct summands generated by projections are well behaved with respect to range orthogonal morphisms. In particular, we show that additive range orthogonality isomorphisms are canonical on proper nonabelian parts of Baer *-algebras. We apply algebraic results to matrix C*-algebras to show that any range orthogonal isomorphisms between them is canonical. The same holds for C*-algebras having proper nonabelian part generated by projections. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper we consider the problem of determining the maximum dimension of P-perpendicular to(A circle plus B)P, where Aand Bare unital, semi-simple subalgebras of the set M-n of n x n complex matrices, and P is an element of M-2n is a projection of rank n. We exhibit a number of equivalent formulations of this problem, including the one which occupies the majority of the paper, namely: determine the minimum dimension of the space A boolean AND S-1BS, where Sis allowed to range over the invertible group GL( n, C) of M-n. This problem in turn is seen to be equivalent to the problem of finding two automorphisms aand beta of M-n for which the dimension of alpha(A) + beta(B) is maximised. It is this phenomenon which gives rise to the title of the paper. (C) 2022 Elsevier Inc. All rights reserved.
Araujo, JoaoBentz, WolframCameron, Peter J.Kinyon, Michael...
30页
查看更多>>摘要：A universal algebra Awith underlying set Ais said to be a matroid algebra if (A, <center dot >), where <center dot > denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping a : X -> A defined on a minimal generating Xof Acan be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. LetAbe any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. In the 1970s, Glazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabo in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabo to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory. (C) 2022 Elsevier Inc. All rights reserved.
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