Aplikasi theorema aliran pada subdigraph untuk menentukan barisan graphical

Barisan Graphical merupakan salah satu aplikasi dan i theorema aliran untuk permasalahan subdigraph. Barisan graphical ini merupakan barisan dari derajat titik sehuah graph. Sebuah graph (p,O) cliperoleh dari sebuah digraph (p,O) dengan suatu transformasi dasar (p,O) d-invarian. Sebuah digraph (p,O) yang ditransfonnasi harus memenuhi kondisi sirkuit ganjil. dan jumlah derajat keluar genap yang sama dengan jumlah derajat masuk dari setiap titiknya. Hasii transformasi ini merupakan sehuah digraph (p,O) simetri yang disebut juga graph(p,0). Suatu barisan dari n integer nonnegatif disebut sebagai barisan graphical jika jumlah dari barisan tersebut adalah genap dan dapat direalisasikan ke dalam sehuah graph. Barisan graphical dapat diubah menjadi barisan dual untuk mempercepat proses penyelesaian graphical (p,O) yang diinginkan dan selanjutnya dapat diubah menjadi barisan modifikasi dual untuk penyelesaian graphical (1,0) yang diinginkan. Graphical sequence is one of applications flows theorem for subdigraph problems. Graphical sequence is the sequence of node degree of graph. Given graph (p,O) is determined from digraph (p,O) with elementary (p,O) d-invariant transformation. Digraph (p,O) which is transformed must be satisfy odd-circuit condition and the addition of outgoing degree is even number that equal incoming degree in every node. The result is digraph (p,O) symetry called graph (p,0). A sequence of n nonnegative integer is graphical sequence if the addition of sequences is even number and. can be realized into a graph. Graphical sequence can be changed into dual sequence to process the graphical (p,0) solution and form modification dual sequence .to determine a graphical (1,0)

The graph bottleneck identity

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$. Five types of matrices that determine transitional measures for a digraph are considered, namely, the matrices of path weights, connection reliabilities, route weights, and the weights of in-forests and out-forests. The results obtained have undirected counterparts. In [P. Chebotarev, A class of graph-geodetic distances generalizing the shortest-path and the resistance distances, Discrete Appl. Math., URL http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic

Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

A Reformulation of Matrix Graph Grammars with Boolean Complexes

Prior publication in the Electronic Journal of Combinatorics.Graph transformation is concerned with the manipulation of graphs by means of rules. Graph grammars have been traditionally studied using techniques from category theory. In previous works, we introduced Matrix Graph Grammars (MGG) as a purely algebraic approach for the study of graph dynamics, based on the representation of simple graphs by means of their adjacency matrices. The observation that, in addition to positive information, a rule implicitly defines negative conditions for its application (edges cannot become dangling, and cannot be added twice as we work with simple digraphs) has led to a representation of graphs as two matrices encoding positive and negative information. Using this representation, we have reformulated the main concepts in MGGs, while we have introduced other new ideas. In particular, we present (i) a new formulation of productions together with an abstraction of them (so called swaps), (ii) the notion of coherence, which checks whether a production sequence can be potentially applied, (iii) the minimal graph enabling the applicability of a sequence, and (iv) the conditions for compatibility of sequences (lack of dangling edges) and G-congruence (whether two sequences have the same minimal initial graph).This work has been partially sponsored by the Spanish Ministry of Science and Innovation, project METEORIC (TIN2008-02081/TIN)