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Reconfiguring Minimum Dominating Sets: The γ-Graph of a Tree
Three open questions about γ- graphs of trees are answered by providing upper bounds on the maximum degree, the diameter, and the number of minimum dominating sets of a graph G. Expand
Minimum number of distinct eigenvalues of graphs
The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are provenExpand
Reconfiguration of Colourings and Dominating Sets in Graphs
Different types of domination reconfiguration graphs are obtained, depending on whether vertices are exchanged along edges or not, when the dominating sets are restricted to be minimum dominating sets. Expand
Totally positive shapes and TPk-completable patterns☆
Abstract The notions of total positivity and of TP k are generalized to “shapes” (a generalization of matrices). In particular, the relationship between positivity of “contiguous” minors and allExpand
Achievable multiplicity partitions in the inverse eigenvalue problem of a graph
Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, andExpand
Note: TP2=Bruhat
It is shown that the Bruhat partial order on permutations is equivalent to a certain natural partial order induced by TP"2 matrices and that a positive matrix is TP"2 if (and only if) it satisfiesExpand
Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]
Abstract We correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3Expand
Complex Hadamard diagonalisable graphs
In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices.Expand
The logarithmic method and the solution to the TP2-completion problem
PAGE A matrix is called TP2 if all 1-by-1 and 2-by-2 minors are positive. A partial matrix is one with some of its entries specified, while the remaining, unspecified, entries are free to be chosen.Expand
Ratio of Matrices