21 problems

Let $\Delta\geq 3$ be an integer. Consider graphs all of whose vertices have degree at most $\Delta$, except possibly one vertex, and let $Z(G;q)$ denote the chromatic polynomial o…

Hawaiian conjecture. Each chord of $\mathcal{G}_{P}$ contains at least one nonreal zero of $P$.

Craven–Csordas–Smith conjecture. The polynomial $W(P,P')$ has at most $2s$ real zeros.

Mean-value formula conjecture. The mean value is equal to

Real-frequency extension conjecture. The Gelfond–Khovanskii formula for the mean value of an exponential sum over the zeros of a system holds for exponential sums with real frequen…

Let ${\cal B}$ be the limiting zero set for a strip of width $L$, let $\#E$ be its number of endpoints, let $\#C$ be its number of connected components, and let $\#Q$ be its number…

The width-two strip conjecture. The family $2_{\rm P}\times n_{\rm F}$ possesses this property for every finite length $n$. The observation at $w=-1/2$ supports the conjecture, but…

For a family of strip lattices, let $f_n(z)$ be the finite-length partition functions and let the limiting zero set be the common limit of their zero sets as the length tends to in…

Sokal's conjecture. Every loopless graph has the multivariate Brown–Colbourn property. This strengthens the univariate Brown–Colbourn conjecture by requiring simultaneous zero-free…

Consider the three-site Potts model on up-pointing triangles, with partition-function variable $w=(e^K-1)/q$. In the thermodynamic limit, let the zeros be viewed in the complex $w$…

Let $q$ be the number of Potts states, let $x=(e^K-1)/\sqrt q$, and let $P_G(q,x)$ be the partition-function polynomial for a finite self-dual square lattice or for a square lattic…

For $q\in(-1,0)$, let $\theta(q,\cdot)$ be the partial theta function, let $x_1(q)$ denote its first positive zero, and consider a conjugate pair of zeros $a\pm ib$ with $a>0$, whe…

Let $B_{5/27}(1)$ denote the disk of radius $5/27$ centered at $1$, and let $Z(G;q)$ be the chromatic polynomial of a series-parallel graph $G$. The punctured-disk zero-free conjec…

Let $P_{n,k}(t)$ be the generating polynomial associated with the Lah distribution. Negative-real-part zeroes conjecture. All complex zeroes of $P_{n,k}$ have negative real parts.…

For each integer $\Delta\geq 3$, let $\mathcal{G}_\Delta$ be the class of graphs of maximum degree at most $\Delta$, and let … where $Z_G(\lambda)$ is the independence polynomial o…

Edge cover polynomial zero-bound conjecture. All complex zeros of $\mathcal E(G,z)$ lie in the open disk

Special-case generalized Hawaii conjecture. The following estimates hold:

Generalized Hawaii conjecture. If $\varkappa>\dfrac{n-1}{n}$, then

Hawaii conjecture. The number of real zeros of $Q_1[p]$ does not exceed the number of non-real zeros of $p$.

Let $G$ be a finite planar self-dual lattice, or let $G$ be the square lattice with free or periodic boundary conditions in the thermodynamic limit. For $q\geq 1$, introduce $v$ by…

Let $q_0(L)$ denote the relevant limiting point for a square-lattice strip of width $L$ with cylindrical boundary conditions, and write $\tau=(1+\sqrt{5})/2$ for the golden ratio.…