Reese, Tyler M.Fehribach, Joseph D.Paffenroth, Randy C.
18页
查看更多>>摘要：The relationship between Kirchhoff graphs and equitable edge partitions of their corresponding digraphs is discussed. It is shown that if the natural edge partition for the associated digraph to a vector graph is equitable and the quotient matrix based on this partition is symmetric, then the vector graph is Kirchhoff. The converse is not true: many Kirchhoff graphs have natural edge partitions that are not equitable. In addition, it is shown that for a digraph with an equitable edge partition, the partition is uniform if and only if the quotient matrix is symmetric. Hence every uniform equitable edge partition of a digraph is a Kirchhoff partition and can generate a Kirchhoff graph.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The present paper considers the operator pencil A(lambda) = A(0) + A(1)lambda, where A(0), A(1) &NOTEQUexpressionL; 0 are bounded linear mappings between complex Hilbert spaces and A0 is neither one-toone nor onto. Assuming that 0 is an isolated singularity of A(lambda) and that the image of A(0) is closed, certain operators are defined recursively starting from A(0) and A(1) and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A(lambda) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A(0) being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A(lambda) at 0. (c) 2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier