scholarly journals Flow Polynomials as Feynman Amplitudes and their $\alpha$-Representation

10.37236/6396 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Andrey Kuptsov ◽  
Eduard Lerner ◽  
Sofya Mukhamedjanova
Keyword(s):  
Finite Field ◽  
Linear Combination ◽  
Connected Graph ◽  
Fourier Transformation ◽  
Spanning Trees ◽  
Standard Technique ◽  
Flow Polynomial ◽  
Feynman Amplitudes

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning  trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)}  \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. Kontsevich  conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by  Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$  is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q \right)^{E(G)}$, and $|V(H)|$ is odd.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review

2018 ◽  
Vol 5 (3) ◽  
pp. 265-281 ◽  
Author(s):  
Maumita Chakraborty ◽  
Sumon Chowdhury ◽  
Joymallya Chakraborty ◽  
Ranjan Mehera ◽  
Rajat Kumar Pal
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Extensive Review

CONSTRUCTING MULTIDIMENSIONAL SPANNER GRAPHS

1991 ◽  
Vol 01 (02) ◽  
pp. 99-107 ◽  
Author(s):  
JEFFERY S. SALOWE
Keyword(s):  
Shortest Path ◽  
Complete Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Input Parameter ◽  
Minimum Spanning Trees ◽  
Positive Edge ◽  
Spanning Subgraph ◽  
Vertex Set

Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in [Formula: see text] and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing [Formula: see text] edges in [Formula: see text] time. We apply this spanner to the construction of approximate minimum spanning trees.

scholarly journals Maximizing the Number of Spanning Trees in a Connected Graph

2020 ◽  
Vol 66 (2) ◽  
pp. 1248-1260
Author(s):  
Huan Li ◽  
Stacy Patterson ◽  
Yuhao Yi ◽  
Zhongzhi Zhang
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Number Of Spanning Trees

scholarly journals On end-faithful spanning trees in infinite graphs

1990 ◽  
Vol 107 (3) ◽  
pp. 461-473 ◽  
Author(s):  
Reinhard Diestel
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Infinite Graphs ◽  
Connected Subgraph ◽  
Initial Vertex ◽  
Infinite Path ◽  
Infinite Component

Let G be an infinite connected graph. A ray (from ν) in G is a 1-way infinite path in G (with initial vertex ν). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X.

Generation of All Spanning Trees of a Simple, Symmetric Connected Graph

2009 ◽  
Author(s):  
Krishnendu Basuli ◽  
Samar Sen Sarma ◽  
Saptarshi Naskar
Keyword(s):  
Connected Graph ◽  
Spanning Trees

scholarly journals Spanning Simplicial Complexes of Uni-Cyclic Graphs

2015 ◽  
Vol 22 (04) ◽  
pp. 707-710 ◽  
Author(s):  
Imran Anwar ◽  
Zahid Raza ◽  
Agha Kashif
Keyword(s):  
Simplicial Complex ◽  
Connected Graph ◽  
Spanning Trees ◽  
Hilbert Series ◽  
Simplicial Complexes ◽  
Cyclic Graphs ◽  
Cyclic Graph

In this paper, we introduce the concept of the spanning simplicial complex Δs(G) associated to a simple finite connected graph G. We characterize all spanning trees of the uni-cyclic graph Un,m. In particular, we give a formula for computing the Hilbert series and h-vector of the Stanley-Reisner ring k[Δs(Un,m)]. Finally, we prove that the spanning simplicial complex Δs(Un,m) is shifted and hence is shellable.

A Contribution to the Theory of Chromatic Polynomials

1954 ◽  
Vol 6 ◽  
pp. 80-91 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Chromatic Polynomials ◽  
Colour Problem ◽  
General Graphs

SummaryTwo polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G, n). Attention is called to some unsolved problems concerning ϕ(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ(G, x, y) in two variables x and y, which can be regarded as generalizing both θ(G, n) and ϕ(G, n) is studied. For a connected graph χ(G, x, y) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.

On the Kirchhoff Index of Graphs

2013 ◽  
Vol 68 (8-9) ◽  
pp. 531-538 ◽  
Author(s):  
Kinkar C. Das
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Type Result ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Number Of Spanning Trees

Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ ... ≥ μn-1 > mn = 0. The Kirchhoff index of G is defined as [xxx] In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus-Gaddum-type result for the Kirchhoff index.

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