On the representability of totally unimodular matrices on bidirected graphs (original) (raw)

A bidirected generalization of network matrices

Networks, 2006

We define binet matrices, which furnish a direct generalization of totally unimodular network matrices and arise from the node-edge incidence matrices of bidirected graphs in the same way as the network matrices do from directed graphs. We develop the necessary theory, give binet representations for interesting sets of matrices, characterize totally unimodular binet matrices and discuss the recognition problem. We also prove that binet matrices guarantee half-integral optimal solutions to linear programs.

Partitions in Matrices and Graphs

European Journal of Combinatorics, 1991

This paper introduces a generalization of association schemes, for arbitrary finite graphs, even for arbitrary square matrices. This serves several ends: it can help to find eigenvalues, it can decide whether a given graph is the graph of an association scheme, it generalizes the notion of tactical decomposition (in both graphs and designs), and it includes coherent configurations as well.

Uncovering generalized-network structure in matrices

Discrete Applied Mathematics, 1993

A generalized-network matrix is a matrix that has at most two nonzeros per column. The generalized-network recognition problem for an arbitrary matrix A is the problem of determining a nonsingular matrix T, if one exists, such that T.4 is a generalized-network matrix. This paper presents a polynomial-time algorithm that under an assumption on the combinatorial structure of A solves the generalized-network recognition problem. A class of matroids called bicircular matroids play an important role in the development of the algorithm.

A note on mixed graphs and matroids

A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.

0. Graph Matrices

Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a large or complicated graph than a pictorial representation. As computers are more adept at manipulating numbers than at recognising pictures, it is standard practice to communicate the specification of a graph to a computer in matrix form. In this chapter, we study various types of matrices associated with a graph, and our study is based on Narsing Deo [63], Foulds [82], Harary [104] and Parthasarathy [180].

A new construction method for networks from matroids

2009

We study the problem of information flow in communication networks with noiseless links in which the dependency relations among the data flowing on the different network edges satisfy matroidal constraints. We present a construction that maps any given matroid to a network that admits vector linear network codes over a certain field if and only if the matroid has a multilinear representation over the same field. This new construction strengthens previous results in the literature and, thus, establishes a deeper connection between network coding and matroid theory. We also explore another, more general, mathematical construct referred to as FD-relation which is more suitable than matroids in capturing the dependency relations in general networks.

A new matrix representation of multidigraphs

AKCE International Journal of Graphs and Combinatorics, 2019

In this article, we introduce a new matrix associated with a multidigraph, named as the complex adjacency matrix. We study the spectral properties of bipartite multidigraphs corresponding to the complex adjacency matrix. It is well known that a simple undirected graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric about the origin (with multiplicity). We show that the result is not true in general for multidigraphs and supply a class of non-bipartite multidigraphs which have this property. We describe the complete spectrum of a multi-directed tree in terms of the spectrum of the corresponding modular tree. As a consequence, we get a class of Hermitian matrices for which the spectrum of a matrix in the class and the spectrum of the modulus (entrywise) of the matrix are the same. c

An introduction to coding sequences of graphs

2016

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over Z_2 which has the consecutive 1's property (i.e., 1's are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over Z_2 which satisfy consecutive 1's property. The set of such vectors is called a coding sequence of...

On algebras and matroids associated to undirected graphs

arXiv (Cornell University), 2020

In this short note we make a few remarks on a class of generalized incidence matrices whose matroids do not depend on the orientation of the underlying graph and natural commutative algebras associated to such matrices.