12 problems

Let $G(n;k_{1},\dots,k_{n})$ be the super edge-magic total unicyclic $(p,q)$-graph consisting of an odd cycle … and $k_i$ pendant vertices adjacent to each $a_i$, for…

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the odd length of its unique cycle. Unicyclic graph square-energy conjecture. … The conjecture is motivated by an exact ca…

Let $G$ be a unicyclic graph on $n$ vertices, and let $\mathfrak{c} \in \mathbb{Z}_{>0}^n$. Write $\operatorname{Ker}(\pi_G^{\mathfrak{c}})$ for the associated toric ideal, and cal…

Let $G_1$ and $G_2$ be connected unicyclic graphs, where either each has a 3-cycle and an odd number of vertices, or each has a $c$-cycle for some $c\geq4$. Let $\mathbf{X}_G$ deno…

Let $D=(d_1,d_2,\dots,d_{t-1},d_t,1,\dots,1)$ be a degree sequence with $d_t\geq 2$ and $d_2\neq 2$, and let $G$ be a unicyclic graph with degree sequence $D$. Define … and let…

Regularity conjecture for t-path ideals. The displayed formula should hold. In particular,

Let $G$ be a unicyclic graph of order $n$ whose cycle has size $k$. The path energy $PE(G)$ is defined for such graphs. Path energy conjecture for unicyclic graphs. $PE(G)$ depends…

Let $\pi=(d_0,d_1,\ldots,d_{k-1},1,\ldots,1)$ be a graphic unicyclic degree sequence with $2\le d_0\le d_1\le\ldots\le d_{k-1}$ and $d_k=\ldots=d_{n-1}=1$. Let $\mathcal{U}_{\pi}$…

Let $P_n$ be a path, $C_n$ a cycle, and let $P_n^6$ be the graph obtained by connecting a vertex of $C_6$ to a terminal vertex of $P_{n-6}$. Among all unicyclic graphs on $n$ verti…

Ilić and Ilić's conjecture. Among all $n$-vertex unicyclic graphs, the graph $U_{n,l}^{3,0}$ has the minimum Laplacian coefficients $c_k$, $k=0,\dots,n$; equivalently,…

Let $G$ be a unicyclic graph and let $T$ be a spanning tree of $G$. The spanning-tree upper-bound conjecture. If $T$ is a spanning tree of a unicyclic graph $G$, then … This conjec…

Caporossi–Cvetković–Gutman–Hansen conjecture. Among all unicyclic graphs on $n$ vertices, the cycle $C_n$ has maximal energy if $n\leq 7$ or $n\in\{9,10,11,13,15\}$. For all other…