Inequalities for Doubly Nonnegative Functions
Copyright policy )Inequalities for Doubly Nonnegative Functions
Let $g$ be a bounded symmetric measurable nonnegative function on $[0,1]^2$, and $\left\lVert g \right\rVert = \int_{[0,1]^2} g(x,y) dx dy$. For a graph $G$ with vertices $\{v_1,v_2,\ldots,v_n\}$ and edge set $E(G)$, we define \[ t(G,g) \; = \; \int_{[0,1]^n} \prod_{\{v_i,v_j\} \in E(G)} g(x_i,x_j) \: dx_1 dx_2 \cdots dx_n \; .\] We conjecture that $t(G,g) \geq \left\lVert g \right\rVert^{|E(G)|}$ holds for any graph $G$ and any function $g$ with nonnegative spectrum. We prove this conjecture for various graphs $G$, including complete graphs, unicyclic and bicyclic graphs, as well as graphs with $5$ vertices or less.
- Smithsonian Tropical Research Institute Panama
- Barro Colorado Island Panama
05C35, 05C22, 26D20, Spectral Theory of Differential Operators, Study of Finite Groups and Graphs, Mathematical analysis, Graph, Signed and weighted graphs, Bounded function, FOS: Mathematics, bounded symmetric measurable nonnegative function, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Conjecture, Mathematical Physics, Other analytical inequalities, Extremal problems in graph theory, Symmetric functions and generalizations, Graph Spectra and Topological Indices, Distance-Regular Graphs, Discrete mathematics, Combinatorics, Physical Sciences, Combinatorics (math.CO), Geometry and Topology, Graphs, Mathematics
05C35, 05C22, 26D20, Spectral Theory of Differential Operators, Study of Finite Groups and Graphs, Mathematical analysis, Graph, Signed and weighted graphs, Bounded function, FOS: Mathematics, bounded symmetric measurable nonnegative function, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Conjecture, Mathematical Physics, Other analytical inequalities, Extremal problems in graph theory, Symmetric functions and generalizations, Graph Spectra and Topological Indices, Distance-Regular Graphs, Discrete mathematics, Combinatorics, Physical Sciences, Combinatorics (math.CO), Geometry and Topology, Graphs, Mathematics
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