Journal ArticleDOI
On vector spaces associated with a graph
TLDR
In this paper, a vector space and its dual are associated with a nonoriented graph and the vectors of each subspace share the topological properties of one of the four entities singled out.Abstract:
Four topological entities are necessary for a complete description of a network graph, as required by orthogonal network theory: the seg, the circ and two others, introduced in this paper, as complementary to the seg and the circ respectively. A vector space and its dual are associated with a nonoriented graph. Both spaces are decomposed in two complementary subspaces. The vectors of each subspace share the topological properties of one of the four entities singled out. The results obtained are extended to a directed graph by associating two dual Z-modules with it. The given description of the network graph is perfectly functional to the orthogonal network theory.read more
Citations
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Journal ArticleDOI
Algebraic Potential Theory on Graphs
TL;DR: A motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network, have been discussed in this paper.
Proceedings ArticleDOI
Greedy optimal homotopy and homology generators
Jeff Erickson,Kim Whittlesey +1 more
TL;DR: It is shown that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm.
Posted Content
Harmonic morphisms and hyperelliptic graphs
Matthew Baker,Serguei Norine +1 more
TL;DR: In this paper, a graph-theoretic analogue of the classical Riemann-Hurwitz formula, called the harmonic morphism of graphs, has been proposed and studied.
Journal ArticleDOI
Relevant cycles in chemical reaction networks
Petra M. Gleiss,Peter F. Stadler,Peter F. Stadler,Andreas Wagner,Andreas Wagner,David A. Fell +5 more
TL;DR: The distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution are characterized to propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood of oscillations in response to external perturbations.
Journal ArticleDOI
Minimal Cycle Bases of Outerplanar Graphs
Josef Leydold,Peter F. Stadler +1 more
TL;DR: Two-connected outerplanar graphs have a unique minimal cycle basis with length $2|E|-|V|$.
References
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Journal ArticleDOI
Matrix Generalizations of Some Theorems on Trees, Cycles and Cocycles in Graphs
TL;DR: In this paper, the authors extend graph theory to arbitrary matrices and give four theorems of graph theory, one about projections onto the cycle and cocycle spaces, another about the intersection of these spaces, and two matrix-tree-theorems.
Journal ArticleDOI
On Vector Spaces Associated with a Graph
TL;DR: In this article, it was shown that the intersection of cut sets, circuits, and subgraphs is the null space of the matrix whose rows correspond to the bases of the cutset bases of $V({\bf Q}) and $V(G) if and only if G has an even number of forests.
Journal ArticleDOI
The Decomposition of a Graph and the Introduction of a New Class of Subgraphs
T. W. Williams,L. M. Maxwell +1 more
Journal ArticleDOI
Algebraic Model of Electrical Network
TL;DR: The set of all possible electrical states of a network is modeled using linear algebra and group theory, and the classification of networks is proposed as far as topological structures and methods of solution are concerned.
Journal ArticleDOI
Topological interpretation of the inverse connection matrix A.sb
Pasquale Pugliese,F. Vacca +1 more
TL;DR: In this paper, a new algebraic model of electrical networks is proposed, by a redefinition of f-cut sets, to give an interpretation in terms of linear graph theory to the inverse connection matrix A.s b, i.e., the transformation matrix between voltages in the orthogonal and the primitive reference frames, for a general choice of independent contours.