Related papers: Counting Flows of $b$-compatible Graphs

For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph…

Let $G$ be a simple, connected graph on $n$ vertices, and further assume that $G$ has disjoint cycles. Let $h$ be a real symmetric matrix supported on $G$ (for example, a discrete Schr\"odinger operator). The eigenvalues of $h$ are ordered…

We study polytopes defined by inequalities of the form $\sum_{i\in I} z_{i}\leq 1$ for $I\subseteq [d]$ and nonnegative $z_i$ where the inequalities can be reordered into a matrix inequality involving a column-convex $\{0,1\}$-matrix. These…

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We…

Let $G$ be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial $F(G,\lambda)$ of $G$ are integers if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether a…

Let $r \geq 2$ be a real number. A complex nowhere-zero $r$-flow on a graph $G$ is an orientation of $G$ together with an assignment $\varphi\colon E(G)\to \mathbb{C}$ such that, for all $e \in E(G)$, the modulus of the complex number…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…

This paper studies signed graphs with possible outer-edges. We introduce and investigate the chain group, the boundary operator, the co-boundary operator, the flow group, the tension group, the homology group, the cohomology group, with…

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total…

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions…

Given a directed graph $G = (V, E)$ with $n$ vertices, $m$ edges and a designated source vertex $s\in V$, we consider the question of finding a sparse subgraph $H$ of $G$ that preserves the flow from $s$ up to a given threshold $\lambda$…

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…

Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}\alpha_i(G)x^i$, where…

The Blow-up Lemma established by Koml\'os, S\'ark\"ozy, and Szemer\'edi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalisations of this result concerning the…

A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $\phi$ is an $A$-flow in $G$, then let $N(\phi)$ be the nullity of $\phi$, that is, the set of…

If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and…

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J.…

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove…

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a…

The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially…