10 problems

Let $H_\omega=-\Delta+V_\omega$ be the Anderson model on the $d$-dimensional Sierpinski simplex graph $(\mathbb{V},\mathcal E)$ for $d\ge 2$. Anderson localization conjecture. For…

Generic horofunction boundary conjecture. If $S$ is empty, the horofunction boundary is as in the stated theorem: it has exactly two Busemann points and countably many non-Busemann…

Let $G$ be a simple graph, let $t$ be a positive integer, and let $S[G,t]$ denote the generalized Sierpiński gasket graph based on $G$ at level $t$. Write $\chi(G)$ for the chromat…

Laplacian spectrum conjecture. The Laplacian spectrum of $S(n,k)$ consists of the following eigenvalues:

Let $S(n,k)$ be the Sierpiński graph defined on $\{1,\ldots,k\}^n$, and let a graph be type-II when its total chromatic number is $\Delta(G)+2$. Jakovac–Klavžar's conjecture. For e…

Subadditivity-plus-sigma conjecture. For all $n,m,\ell_a,\ell_b\in\mathbb N$ with $m^n\geq\ell_a\geq\ell_b>0$, if $\ell_a+\ell_b\leq m^n$, then

Decorated Sierpinski graph conjecture. For every $\ell$ with $0\leq\ell\leq m^n$,

Let $\hat{S_p^n}$ be the generalized Sierpiński triangle graph with integers $n\geq 3$ and $p\geq 4$, and let $f(G)$ denote the feedback vertex number of a graph $G$, namely the mi…

Let $I=\left\{0,1,\ldots,s-1\right\}$, $J=\left\{s,s+1,\ldots,s+t-1\right\}$, and $K=\left\{s+t,s+t+1,\ldots,m-1\right\}$, where $s,t\geq0$ and $s+t\leq m$. Let $S_{s,t}(n,m)$ be t…

Let $S(n,m)$ be the generalized and expanded Sierpinski graph, with vertex set $V_{S(n,m)}=\left\{0,1,\ldots,m-1\right\}^{n}$. Let $Lex$ denote the lexicographic order on this vert…