What are representations of the minimum degree sum of $k$ nonadjacent vertices? I was looking over the statement of Ore's theorem in Chartrand, Lesniak and Zhang's "Graphs & Digraphs" (p. 129 in the sixth edition) and I noticed they use the notation $\sigma_2 (G)$ for the minimum degree sum of two nonadjacent vertices of a graph $G$. I've never seen this notation anywhere else - are there any other interesting places where this number (or $\sigma_k (G)$, or similar) pops up, maybe in relation to the Laplacian matrix of $G$ or something? Or is this quantity only useful for Ore's theorem and similar statements about Hamiltonicity?
It seems Wikipedia doesn't have anything interesting, and while a quick Google Scholar search shows that degree sums are indeed an area of interest, I couldn't really parse through the results to find exactly what I'm asking for.

Conditions involving a lower bound on $\sigma_2(G)$ are commonly called "Ore-type conditions". We see them mainly in two situations:

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*Conditions for Hamiltonian cycles in which the bound on $\sigma_2(G)$ can be lowered in exchange for some additional assumptions, e.g. in tough graphs or induced $K_{1,t}$-free graphs.

*Conditions for other (often cycle-related) objects to appear, such as $k$-factors or disjoint collections of $k$ cycles.

Often, when there is a "Dirac-type condition" (a lower bound on minimum degree) for some kind of theorem, there is also an Ore-type condition generalizing it.
There are also results assuming lower bounds on $\sigma_k(G)$: the minimum sum of degrees over all $k$-vertex independent sets in $G$. These are less common, because if you're assuming a weirder condition, your results had better be correspondingly more impressive for anyone to be interested. (Also, dealing with the more complicated condition is harder in the first place.)
