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Full-Text Articles in Physical Sciences and Mathematics
Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody
Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody
Undergraduate Honors Theses
A graph has quantum symmetry if the algebra associated with its quantum automorphism group is non-commutative. We study what quantum symmetry means and outline one specific method for determining whether a graph has quantum symmetry, a method that involves studying planar algebras and manipulating planar tangles. Modifying a previously used method, we prove that the 5-cycle has no quantum symmetry by showing it has the generating property.
The Minimum Number Of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph Is A Tree, Wenxuan Ding
The Minimum Number Of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph Is A Tree, Wenxuan Ding
Undergraduate Honors Theses
For a tree T, U(T) denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices whose graph is T. It is known that U(T) >= 2. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. If T′ is a linear tree resulting from the addition of 1 vertex to T, we show that |U(T′)−U(T)|
A Survey Of Methods To Determine Quantum Symmetry Of Graphs, Samantha Phillips
A Survey Of Methods To Determine Quantum Symmetry Of Graphs, Samantha Phillips
Undergraduate Honors Theses
We introduce the theory of quantum symmetry of a graph by starting with quantum permutation groups and classical automorphism groups. We study graphs with and without quantum symmetry to provide a comprehensive view of current techniques used to determine whether a graph has quantum symmetry. Methods provided include specific tools to show commutativity of generators of algebras of quantum automorphism groups of distance-transitive graphs; a theorem that describes why nontrivial, disjoint automorphisms in the automorphism group implies quantum symmetry; and a planar algebra approach to studying symmetry.