Let $\mathcal{H}$ be an $r$-uniform hypergraph. The minimum positive co-degree of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the $k$ such that if $S$ $(r-1)$-set contained in a hyperedge then at least hyperedges $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine maximum possible size intersecting $n$-vertex hypergraph with $\delta_{r-1}^+(\mathcal{H}) \...

We prove a minimum degree version of the Kruskal--Katona theorem for triple systems: given $d\ge 1/4$ and system $\mathcal{F}$ on $n$ vertices with $\delta(\mathcal{F})\ge d\binom n2$, we obtain asymptotically tight lower bounds size its shadow. Equivalently, $t\ge n/2-1$, determine graph vertices, in which every vertex is contained at least $\binom t2$ triangles. This can be viewed as variant ...

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...