5 problems

Let $G$ be an $n$-vertex graph. The weighted Szeged index conjecture. The weighted Szeged index of $G$, denoted by $\operatorname{wSz}(G)$, is at most … Equality is attained only b…

Let $T$ and $T'$ be trees, and let $Sz_k$ denote the Steiner $k$-Szeged index and $Sz$ the classical Szeged index. Conjecture. For any two trees $T$ and $T'$, … if and only if … Th…

Let $G$ be a $2$-connected graph of order $n\ge 10$. Let $K_n$ denote the complete graph, and let $K_n^2$ and $K_n^{n-2}$ be the graphs obtained from $K_{n-1}$ by adding one vertex…

Let $G$ be a connected graph and let $B_1,\ldots,B_k$ be all its non-complete blocks, of respective orders $n_1,\ldots,n_k$. Define the Szeged–Wiener difference by…

Let $G$ be a connected graph with $n \geq 5$ vertices and girth $g\geq 5$, and with an odd cycle. The Hansen–Li–Liu conjecture. asserts that … Moreover, the bound should be best po…