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5 days ago · If G = (V, E) is a simple undirected graph and e ∈ E, let G ∖ e = (V ∖ e, E ∩ [V ∖ e]2). If G = (V, E) is a finite graph, let ω(G) be the size of the largest clique in G and let χ(G) be the chromatic number. Question.
Jun 16, 2024 · For the clique problem, one would use $\varphi = \forall x\forall y((Xx \wedge Xy \wedge \neg x=y) \to Exy)$, where $E$ is the edge predicate. $\varphi$ says that $X$ is a clique. Proof sketch: From $\varphi$, derive a tree automaton. The tree it works on would encode tree decompositions (as for Courcelle's theorem) plus the predicate $X$.
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Jun 30, 2024 · This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the Q-polynomial property.
Jun 29, 2024 · We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times.
Jun 30, 2024 · Let Γ denote a distance-regular graph with diameter D ≥ 2. Let E denote a primitive idempotent of Γ with respect to which Γ is Q -polynomial. Assume that there exists a 3 -clique x, y, z such that Ex^, Ey^, Ez^ are linearly dependent.
Jun 22, 2024 · In this paper, we study the broadcasting problem in a topology that can be represented by the Windmill graph \(Wd_{k,l}\), which contains k cliques of size l, connected to a universal node. We also investigate the broadcast problem in Star of Cliques, which is an extension of the Windmill graph with arbitrary clique sizes. We present an ...
