Combinatorial Matrix Theory

By Leslie Hogben. 2013

527 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b920107592991284db634430d26ab8ff74827222

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In Computational Theory and Mathematics

Citations in the corpus

Minimum number of distinct eigenvalues of graphs (2013, background): “In the context of the (0, 1)-adjacency matrix, A(G), it is well known that q(A(G)) is at least one more than the diameter of G (denoted by diam(G)) (see [4]).”
Consensus in Directed Networks of Agents With Nonlinear Dynamics (2011, background): “Then there exists a positive-definite diagonal matrix , such that is symmetric and , .”
Spectral Graph Theory (2012, unknown): “I also recommend [10, 99, 6, 100, 101].”
Consensus of multi‐agent systems with nonlinear dynamics and sampled‐data information: a delayed‐input approach (2013, background): “A / contains a directed spanning tree [31].”
Normalized graph Laplacians for directed graphs (2011, background): “Using this decomposition, the Laplace operator ∆ can be represented in the Frobenius normal form [6], i.”
Commute Times for a Directed Graph using an Asymmetric Laplacian (2011, background): “In [4] L is called the Formal Laplacian.”
On some properties of three different types of triangular blocked tensors (2016, background): “2 ([2]) Let A be an (n1, · · · , nr) normal upper triangular blocked matrix of order n with diagonal blocks A1, · · · , Ar, and B be an (m1, · · · ,mt) normal upper triangular blocked matrix of order n with diagonal blocks B1, · · · , Bt, respectively.”
Consensus in multi‐agent systems with communication constraints (2012, background): “Lemma 4 (Frobenius normal form (Brualdi and Ryser [38])) There exist a permutation matrix W with order N and an integer m 1, such that”
Synchronization via Pinning Control on General Complex Networks (2013, background): “Note that the condensation network G̃∗ of any directed network G̃ has no closed directed walks [5], where a directed (undirected) walk from vertex vj to vi is a sequence of edges (vi, vi1), (vi1 , vi2), .”
Asymptotic properties of consensus-type algorithms for networked systems with regime-switching topologies (2011, methodology): “Since G contains a spanning tree, by Brualdi and Ryser (1991, Lemma 2.5.1), H has rank n−1. From the expressionM = −H GH , these imply thatM is negative semi-definite and has rank r − 1. (2) By M = −H GH , it can be readily verified that all offdiagonal elements of M are in the form of 0 or gij > 0.”