Related papers: The Number of Nowhere-Zero Flows on Graphs and Sig…
An indecomposable flow $f$ on a signed graph $\Sigma$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge…
The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that…
Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
This article is the complement to [quant-ph/0611284], which proves that flows (as introduced by [quant-ph/0506062]) can be found efficiently for patterns in the one-way measurement model which have non-empty input and output subsystems of…
For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular…
The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex…
While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kahler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely…
Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of computing Kostant partition functions into a…
Given a $2$-$(v,k,\lambda)$ design, $\cal{S}=(X,\cal{B})$, a {\it zero-sum $n$-flow} of $\cal{S}$ is a map $f: \cal{B} \longrightarrow \{\pm 1, \ldots ,\pm (n-1)\}$ such that for any point $x\in X$, the sum of $f$ around all the blocks…
We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr\"odinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive…
Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire…
The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbour colour forcing rule. The zero forcing number provides topological information about linear algebra on graphs,…
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted…
We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…
Let \Gamma be a signed graph and let A(\Gamma) be the adjacency matrix of \Gamma. The nullity of \Gamma is the multiplicity of eigenvalue zero in the spectrum of A(\Gamma). In this paper we characterize the signed graphs of order n with…
In this paper, we study graphs whose matching polynomial have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs.. We show that…
Converting modulo flows into integer-valued flows is one of the most critical steps in the study of integer flows. Tutte and Jaeger's pioneering work shows the equivalence of modulo flows and integer-valued flows for ordinary graphs.…
This paper proves that for any positive integer $k$, every essentially $(2k+1)$-unbalanced $(12k-1)$-edge connected signed graph has circular flow number at most $2+\frac 1k$.
In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic…