Spectral conditions for graph rigidity in the Euclidean plane

Rigidity is the property of a structure that does not flex. It well studied in discrete geometry and mechanics, has applications material science, engineering biological sciences. A bar-and-joint framework pair (G,p) graph G together with map p vertices into Euclidean space. We view edges as bars universal joints. The can move continuously long distances between pairs adjacent are preserved. rigid if any such motion preserves all vertices. In 1970, Laman obtained combinatorial characterization graphs plane. 1982, Lovász Yemini discovered new proved every 6-connected rigid. Combined global rigidity given by Jackson Jordán 2009, it actually globally Consequently, algebraic connectivity greater than 5, then this paper, we improve bound show for minimum degree ??6, its 2+1??1, 2+2??1, Our results imply connected regular Ramanujan at least 8 also prove more general result giving sufficient spectral condition existence k edge-disjoint spanning subgraphs. same implies contains 2-connected This extends previous conditions packing trees.