17 problems

The conjecture asserts that for every matchable nanotube $N$, its resonance graph $R_6(N)$ is disconnected. Here $R_6(N)$ is the graph whose vertices are the perfect matchings of…

Zagreb index comparison conjecture. If $G$ is a connected graph, then

Let $G$ be a graph with $n$ vertices, and let $cM_{2}(G)$ denote its complementary second Zagreb index, defined by … Here $K_m\vee\overline{K_{n-m}}$ is the join of the complete gr…

Branch-type conjecture. For sufficiently large $n$, minimal-ABC trees have only $B_3$-branches besides the big vertices.

Star conjecture. The subgraph of a minimal-ABC tree induced by its big vertices is a star.

Let $B_2(n)$ denote the family of bicyclic graphs specified in the paper, and let $G\in B_2(n)$ have order $n\geq 11$. The Graovac-Ghorbani atom-bond connectivity index is denoted…

Let $\mathcal{B}'_n$ be the family of all bicyclic graphs on $n$ vertices, and let $G\in\mathcal{B}'_n$ have order $n\geq 8$. The Graovac-Ghorbani atom-bond connectivity index is d…

Spectral characterization conjecture. (a) For all feasible $n\ne44$, two $C_n$-isomers are isomorphic if and only if they are cospectral with respect to $T_n$.

Dumbbell-like graph conjecture. Dumbbell-like graphs $D_{a,b,a'}^{\ell}$ attain the minimum value of sum-Balaban index.

Asymptotic dumbbell conjecture. Among all dumbbell graphs $D_{a,b,a'}$ on $n$ vertices, the minimum is achieved for one with

Balanced dumbbell conjecture. Among all dumbbell graphs $D_{a,b,a'}$ on at least $14$ vertices, the minimum value of sum-Balaban index is achieved for one with $a'=a$ or $a'=a+1$.

Extremal tree conjecture. The $n$-vertex tree $T$ has the maximum $AZI$ value if and only if $T$ has the minimum $ABC$ index.

Cao–Dehmer–Schaumann's conjecture. We have

Let $\mathcal{R}(n)$ be the class of trees on $n$ vertices, and let $BP^0$ denote the Basic regression with minimum value $BP^{0*}_\mathcal{R}(n)$ over $\mathcal{R}(n)$. A tree is…

Nine-$B_2$-branch conjecture. A minimal-ABC tree can contain at most nine $B_2$-branches.

Three-$B_1$-branch conjecture. A minimal-ABC tree can contain at most three $B_1$-branches.

Let $G$ be a tree with minimal atom-bond connectivity (ABC) index among all trees of size $n$. The trees $T_0,T_1,T_2,T_3,T_4,T_5,T_6$ are the structures depicted in Figure. Gutman…