20 problems

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. Welsh's conjecture. For any bridgeless graph $G$, … The conjecture parallels the Birkhoff–Lewis resul…

Let $G$ be a cyclically 4-connected cubic graph, and let $F(G,t)$ denote its flow polynomial. Let $\tau=(1+\sqrt{5})/2$. Finiteness conjecture. For every $\epsilon>0$, only finitel…

Let $G$ be a 3-connected graph with $n$ vertices and $m$ edges, and let $F(G,t)$ denote its flow polynomial. Let $\phi\approx1.749$ be the flow root of $K_5$ in $(1,2)$. 3-connecte…

Let $G$ be a cubic bridgeless graph with $E$ edges, let $F_G(Q)$ denote its flow polynomial, and let $\phi=(1+\sqrt{5})/2$ be the golden ratio. Golden inequality conjecture. … More…

Let $F_G(q)$ denote the flow polynomial of a graph $G$. Jackson–Salas' eventual positivity conjecture. There exists $q_0\in\mathbb{R}$ such that … for all $2$-connected graphs $G$…

Conjugate golden identity inequality conjecture. For any cubic bridgeless graph $G$,

Let $G$ be a bridgeless graph, and let the flow roots of $G$ be the roots of its flow polynomial $F(G,\lambda)$. Real flow-root conjecture. If all flow roots of $G$ are real, then…

Let $\mathtt{Frk}_{i,n-i}$ be the fork with $n$ vertices and stem of size $i$, and let $\#\mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)$ denote the number of closed flows of size $k$ on thi…

Consider a family of non-planar connected graphs $G_{n,L}$ with $n$ identical layers of width $L$ and periodic boundary conditions in the $n$-direction, together with a curve…

Consider a family $G_{n,L}$ of non-planar connected graphs as above. Let $Q_\ell$ be the smallest $Q>0$ for which the transfer matrix $\widetilde{\sf T}_{L,\ell,(\ell)}$ for the fu…

Consider a family $G_{n,L}$ of non-planar connected graphs made of $n$ identical layers of width $L$, with periodic boundary conditions in the $n$-direction. Let $\mathcal{A}_L$ an…

Let $G(nk,k)$ be the generalized Petersen graph family and let $Q_c(k)$ be the associated real crossing value. Asymptotic real-flow-root conjecture. There exists a real number…

For a family of generalized Petersen graphs $G(nk,k)$, let $Q_c(k)$ be the real crossing value and let $\mu_{k+1,\ell,(\ell),\star}$ denote the indicated dominant eigenvalues. Domi…

Let $\mathcal{B}_\infty$ be the limiting curve conjecturally obtained from the curves $\mathcal{B}_k$ after removing their outward branches. Density conjecture. The set of accumula…

For each $k\geq1$, let $\mathcal{B}_k$ be the limiting curve of flow-polynomial zeros for $G(nk,k)$, with its outward branches removed. Limiting-curve convergence conjecture. As…

For fixed $k\geq1$, let $\mathcal{B}_k$ be the limiting curve of non-isolated accumulation points of zeros of $\Phi_{G(nk,k)}$ as $n\to\infty$. Let $\theta_{k,j}$ denote the asympt…

For $1\leq k\leq3$, consider the flow polynomials of the generalized Petersen graphs $G(nk,k)$ as $n$ varies. Low-width real-root conjecture. The fact that all computed flow roots…

Let $G(nk,k)$ be a generalized Petersen graph, and let its flow polynomial be expressed through the transfer-matrix eigenvalues appearing in the complete decomposition for…

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. The six-flow conjecture. For any bridgeless graph $G$, … The source presents this as a weaker conject…

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. Haggard–Pearce–Royle conjecture. For any bridgeless graph $G$, … This is a proposed weakening of Wels…