23 problems

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…

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…

Let $G$ be a plane graph, and let a largest Clar-set mean a Clar-set of maximum cardinality, while a largest Fries-set means a Fries-set of maximum cardinality. Clar–Fries extensio…

Let $G=(S,T;E)$ be a 2-connected perfectly matchable bipartite plane graph. Let $F$ be its node–inner-face incidence matrix and $Q$ its standard node–edge incidence matrix, and def…

Let $G$ be a convex benzenoid, let $\operatorname{\mathfrak{a}}(G,H)$ be its altan for an attachment set $H$, and let $\eta$ and $\eta(\operatorname{\mathfrak{a}})$ denote the null…

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.

Structural conjecture. For sufficiently large orders, a minimal-ABC tree is comprised solely of a root vertex and $D_z$- and $D_{z+1}$-branches.

Let a fullerene graph be a graph of an arbitrary order, and let its Wiener index be the sum of graph distances over all unordered pairs of vertices. A nanotubical fullerene graph i…

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…

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 $T_n$.

Chemical-stability conjecture. The family of icosahedral fullerenes has the highest chemical stability among all fullerenes with $n$ atoms.

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…

Structural conjecture for large minimal-ABC trees. Enough large minimal-ABC trees are comprised exclusively of $B_3$-branches and internal vertices.

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…