Browsing by Author "Brimkov, Boris"

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    Efficient Computation of Chromatic and Flow Polynomials
    (2015-10-13) Brimkov, Boris; Hicks, Illya V
    This thesis surveys chromatic and flow polynomials, and presents new efficient methods to compute these polynomials on specific families of graphs. The chromatic and flow polynomials of a graph count the number of ways to color and assign flow to the graph; they also contain other important information such as the graph's chromatic number, Hamiltonicity, and number of acyclic orientations. Unfortunately, these graph polynomials are generally difficult to compute; thus, research in this area often focuses on exploiting the structure of specific families of graphs in order to characterize their chromatic and flow polynomials. In this thesis, I present closed formulas and polynomial-time algorithms for computing the chromatic polynomials of novel generalizations of trees, cliques, and cycles; I also use graph duality to compute the flow polynomials of outerplanar graphs and generalized wheel graphs. The proposed methods are validated by computational results.
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    Graph Coloring, Zero Forcing, and Related Problems
    (2017-05) Brimkov, Boris
    This thesis investigates several problems related to classical and dynamic coloring of graphs, and enumeration of graph attributes. In the first part of the thesis, I present new efficient methods to compute the chromatic and flow polynomials of specific families of graphs. The chromatic and ow polynomials count the number of ways to color and assign ow to the graph, and encode information relevant to various physical applications. The second part of the thesis focuses on zero forcing-- a dynamic graph coloring process whereby at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. Zero forcing has applications in linear algebra, quantum control, and power network monitoring. A connected forcing set is a connected set of initially colored vertices which forces the entire graph to become colored; the connected forcing number is the cardinality of the smallest connected forcing set. I present a variety of structural results about connected forcing, such as the effects of vertex and edge operations on the connected forcing number, the relations between the connected forcing number and other graph parameters, and the computational complexity of connected forcing. I also give efficient algorithms for computing the connected forcing numbers of different families of graphs, and characterize the graphs with extremal connected forcing numbers. Finally, I investigate several enumeration problems associated with zero forcing, such as the exponential growth of certain families of forcing sets, relations of families of forcing sets to matroids and greedoids, and polynomials which count the number of distinct forcing sets of a given size.
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    Graph Coloring, Zero Forcing, and Related Problems
    (2017-08-09) Brimkov, Boris; Hicks, Illya V
    This thesis investigates several problems related to classical and dynamic coloring of graphs, and enumeration of graph attributes. In the first part of the thesis, I present new efficient methods to compute the chromatic and flow polynomials of specific families of graphs. The chromatic and flow polynomials count the number of ways to color and assign flow to the graph, and encode information relevant to various physical applications. The second part of the thesis focuses on zero forcing - a dynamic graph coloring process whereby at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. Zero forcing has applications in linear algebra, quantum control, and power network monitoring. A connected forcing set is a connected set of initially colored vertices which forces the entire graph to become colored; the connected forcing number is the cardinality of the smallest connected forcing set. I present a variety of structural results about connected forcing, such as the effects of vertex and edge operations on the connected forcing number, the relations between the connected forcing number and other graph parameters, and the computational complexity of connected forcing. I also give efficient algorithms for computing the connected forcing numbers of different families of graphs, and characterize the graphs with extremal connected forcing numbers. Finally, I investigate several enumeration problems associated with zero forcing, such as the exponential growth of certain families of forcing sets, relations of families of forcing sets to matroids and greedoids, and polynomials which count the number of distinct forcing sets of a given size.