In this paper we show that every family of triples, that is, a 3-uniform hypergraph, with minimum degree at least [...] contains a tight Hamiltonian cycle

In this note we consider a discrete symmetric function f(x, y) where $$f(x,a)+f(y,b)⩾f(y,a)+f(x,b)foranyx⩾yanda⩾b,$$ associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as $$\sum _{uv\in E\left(T\right)}f\left(deg\right(u),deg(v\left)\right),$$ are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randic index follow as corollaries. The extremal structures...