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Full-Text Articles in Discrete Mathematics and Combinatorics
Modern Theory Of Copositive Matrices, Yuqiao Li
Modern Theory Of Copositive Matrices, Yuqiao Li
Undergraduate Honors Theses
Copositivity is a generalization of positive semidefiniteness. It has applications in theoretical economics, operations research, and statistics. An $n$-by-$n$ real, symmetric matrix $A$ is copositive (CoP) if $x^T Ax \ge 0$ for any nonnegative vector $x \ge 0.$ The set of all CoP matrices forms a convex cone. A CoP matrix is ordinary if it can be written as the sum of a positive semidefinite (PSD) matrix and a symmetric nonnegative (sN) matrix. When $n < 5,$ all CoP matrices are ordinary. However, recognizing whether a given CoP matrix is ordinary and determining an ordinary decomposition (PSD + sN) is still an unsolved problem. Here, we give an overview on modern theory of CoP matrices, talk about our progress on the ordinary recognition and decomposition problem, and emphasis the graph theory aspect of ordinary CoP matrices.
Enumerating Switching Isomorphism Classes Of Signed Graphs, Nathaniel Healy
Enumerating Switching Isomorphism Classes Of Signed Graphs, Nathaniel Healy
Undergraduate Honors Theses
Let Γ be a simple connected graph, and let {+,−}^E(Γ) be the set of signatures of Γ. For σ a signature of Γ, we call the pair Σ = (Γ,σ) a signed graph of Γ. We may define switching functions ζ_X ∈ {+, −}^V (Γ) that negate the sign of every edge {u, v} incident with exactly one vertex in the fiber X = ζ^{−1}(−). The group Sw(Γ) of switching functions acts X on the set of signed graphs of Γ and induces an equivalence relation of switching classes in its orbits; there are 2^{|E(Γ)|−|V (Γ)|+1} such classes. More interestingly, …
The Enumeration Of Minimum Path Covers Of Trees, Merielyn Sher
The Enumeration Of Minimum Path Covers Of Trees, Merielyn Sher
Undergraduate Honors Theses
A path cover of a tree T is a collection of induced paths of T that are vertex disjoint and cover all the vertices of T. A minimum path cover (MPC) of T is a path cover with the minimum possible number of paths, and that minimum number is called the path cover number of T. A tree can have just one or several MPC's. Prior results have established equality between the path cover number of a tree T and the largest possible multiplicity of an eigenvalue that can occur in a symmetric matrix whose graph is that tree. We …