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Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices.
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This content will become publicly available on March 1, 2026
The k -core in percolated dense graph sequences
Abstract We determine the order of thek-core in a large class of dense graph sequences. Let$$G_n$$be a sequence of undirected,n-vertex graphs with edge weights$$\{a^n_{i,j}\}_{i,j \in [n]}$$that converges to a graphon$$W\colon[0,1]^2 \to [0,+\infty)$$in the cut metric. Keeping an edge (i,j) of$$G_n$$with probability$${a^n_{i,j}}/{n}$$independently, we obtain a sequence of random graphs$$G_n({1}/{n})$$. Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of thek-core of random graphs$$G_n({1}/{n})$$. Our result can also be used to obtain the threshold of appearance of ak-core of ordern.
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- Award ID(s):
- 2106556
- PAR ID:
- 10627214
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Applied Probability
- Volume:
- 62
- Issue:
- 1
- ISSN:
- 0021-9002
- Page Range / eLocation ID:
- 298 to 318
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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