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The inertia of unicyclic graphs and bicyclic graphs

100%

Liu Y.

Discussiones Mathematicae - General Algebra and Applications

2013

tom 33

nr 1

109-115

Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications, 431 (2009), 1293-1301.] proved if G is a unicyclic graph, then η(G) equals n - 2ν(G) - 1, n-2ν(G) or n - 2ν(G) + 2. Barrett et al. determined the inertia sets for trees and graphs with cut vertices. In this paper, we give the nullity of bicyclic graphs 𝓑ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and 𝓑ₙ⁺⁺, respectively.

Matrix quadratic equations column/row reduced factorizations and an inertia theorem for matrix polynomials

86%

Karelin I., Lerer L.

International Journal of Applied Mathematics and Computer Science

2001

tom 11

nr 6

1285-1310

It is shown that a certain Bezout operator provides a bijective correspondence between the solutions of the matrix quadratic equation and factorizatons of a certain matrix polynomial (which is a specification of a Popov-type function) into a product of row and column reduced polynomials. Special attention is paid to the symmetric case, i.e. to the Algebraic Riccati Equation. In particular, it is shown that extremal solutions of such equations correspond to spectral factorizations of . The proof of these results depends heavily on a new inertia theorem for matrix polynomials which is also one of the main results in this paper.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 International Journal of Applied Mathematics and Computer Science

1 Karelin I.

1 Lerer L.

1 Liu Y.

1 2013

1 2001

Wyniki wyszukiwania

The inertia of unicyclic graphs and bicyclic graphs

Matrix quadratic equations column/row reduced factorizations and an inertia theorem for matrix polynomials