13 problems

Let $G=(V,E)$ be a connected multigraph, and let $\operatorname{Rel}(G;q)$ be its all-terminal reliability polynomial. Define … Let $\overline{\mathbb{D}}$ denote the closed unit d…

Let $G$ be a loopless connected graph, let $R_G({\bf p})$ be its multivariate reliability polynomial, and let $C_G({\bf v})$ be the connected-spanning-subgraph polynomial related b…

Let $\mathrm{M}$ be a matroid, and define $h_i(\mathrm{M})$ by … where $f_i(\mathrm{M})$ counts independent subsets of size $i$. Dawson's conjecture. For any $i$ and any matroid…

Let $G$ be a connected graph, and let $R_G(p)$ be its univariate reliability polynomial, obtained by setting all edge-operation probabilities equal to $p$. Univariate Brown–Colbour…

Let $G=(V,E)$ be a connected simple graph, and let $\operatorname{Rel}(G;q)$ be its all-terminal reliability polynomial. Define … For a subset of $\mathbb{C}$, write an overline fo…

Amicability conjecture. For every network $G$ and every pair of distinct vertices $v$ and $w$, the pair $\{v,w\}$ is amicable in $G$.

Let $G$ be a connected graph, and let $R(G,p)$ denote its reliability polynomial. Brown–Colbourn disk conjecture. Every root $p$ of $R(G,p)$ lies in the closed disk … The conjectur…

Generalized connected-sum monotonicity conjecture. If $|1+v_e|\leq 1$ for every edge $e$, then $|C_G(\lambda,\{v_e\})|$ is a decreasing function of $\lambda$ on…

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

Let $G$ range over finite connected graphs, and consider the real roots of their reliability polynomials, including the always-present root $q=1$. The closure of the set of real re…

For a graph $G$, consider a maximal interval of decrease of its node reliability polynomial $\operatorname{nRel}(G;p)$ within $(0,1)$. The empty graph $\overline{K}_n$ has a decrea…

Let $G$ be a graph with at least two cut vertices, and let $\operatorname{nRel}(G;p)$ denote its node reliability polynomial. A fixed point is a value $p\in(0,1)$ satisfying…

Let $T$ be a tree of order $n\geq 4$, and let its node reliability polynomial be denoted by $\operatorname{nRel}(T;p)$. An inflection point is a point in $(0,1)$ where the concavit…