Publicação
The number of P-vertices in a matrix with maximum nullity
| Resumo: | Let T be a tree with n >= 2 vertices. Set 8(T) for the set of all real symmetric matrices whose graph is T. Let A is an element of S(T) and i is an element of {1, . . . , n}. We denote by A(i) the principal submatrix of A obtained after deleting the row and column i. We set m(A) (0) for the multiplicity of the eigenvalue zero in A (the nullity of A). When m(A(i)) (0) = m(A) (0) + 1, we say that i is a P-vertex of A. As usual, M(T) denotes the maximum nullity occurring of B is an element of S(T). In this paper we determine an upper bound and a lower bound for the number of P-vertices in a matrix A is an element of S(T) with nullity M(T). We also prove that if the integer b is between these two bounds, then there is a matrix E is an element of S(T) with b P-vertices and maximum nullity. (C) 2018 Elsevier Inc. All rights reserved. |
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| Autores principais: | Fernandes, Maria do Rosário Silva Franco |
| Outros Autores: | da Cruz, Henrique F. |
| Assunto: | Trees Acyclic matrices Maximum nullity Parter vertices |
| Ano: | 2018 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade Nova de Lisboa |
| Idioma: | inglês |
| Origem: | Repositório Institucional da UNL |
| Resumo: | Let T be a tree with n >= 2 vertices. Set 8(T) for the set of all real symmetric matrices whose graph is T. Let A is an element of S(T) and i is an element of {1, . . . , n}. We denote by A(i) the principal submatrix of A obtained after deleting the row and column i. We set m(A) (0) for the multiplicity of the eigenvalue zero in A (the nullity of A). When m(A(i)) (0) = m(A) (0) + 1, we say that i is a P-vertex of A. As usual, M(T) denotes the maximum nullity occurring of B is an element of S(T). In this paper we determine an upper bound and a lower bound for the number of P-vertices in a matrix A is an element of S(T) with nullity M(T). We also prove that if the integer b is between these two bounds, then there is a matrix E is an element of S(T) with b P-vertices and maximum nullity. (C) 2018 Elsevier Inc. All rights reserved. |
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