Matchings in infinite graphs
Matrices and graphs in Euclidean geometry.
Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs
In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...
Matrices representable by directed graphs
Matrix and discrepancy view of generalized random and quasirandom graphs
We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree...
Matrix inversion and digraphs: the one factor case.
Matrix partitions with finitely many obstructions.
Matrix-free proof of a regularity characterization.
Matroidal decomposition of a graph
Matroids in terms of Cayley graphs and some related results.
Matroids over a ring
We introduce the notion of a matroid over a commutative ring , assigning to every subset of the ground set an -module according to some axioms. When is a field, we recover matroids. When , and when is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and...
Maxclique and Unit Disk Characterizations of Strongly Chordal Graphs
Maxcliques (maximal complete subgraphs) and unit disks (closed neighborhoods of vertices) sometime play almost interchangeable roles in graph theory. For instance, interchanging them makes two existing characterizations of chordal graphs into two new characterizations. More intriguingly, these characterizations of chordal graphs can be naturally strengthened to new characterizations of strongly chordal graphs
Maximal buttonings of trees
A buttoning of a tree that has vertices v1, v2, . . . , vn is a closed walk that starts at v1 and travels along the shortest path in the tree to v2, and then along the shortest path to v3, and so forth, finishing with the shortest path from vn to v1. Inspired by a problem about buttoning a shirt inefficiently, we determine the maximum length of buttonings of trees
Maximal Canonical Graphs with Seven Nonzero Eigenvalues
Maximal Canonical Graphs with Three Negative Eigenvalues
Maximal flat antichains of minimum weight.
Maximal Graphs with Given Connectivity and Edge-Connectivity
Maximal graphs with respect to hereditary properties
A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property ; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P...
Maximal hypergraphs with respect to the bounded cost hereditary property
The hereditary property of hypergraphs generated by the cost colouring notion is considered in the paper. First, we characterize all maximal graphs with respect to this property. Second, we give the generating function for the sequence describing the number of such graphs with the numbered order. Finally, we construct a maximal hypergraph for each admissible number of vertices showing some density property. All results can be applied to the problem of information storage.
Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC
In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles.