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Characterization of Linearly Separable Boolean Functions: A Graph-Theoretic Perspective.

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    This study introduces a graph-theoretic method to analyze Boolean functions. Linearly separable Boolean functions are shown to correspond to a new graph structure called hyperstars.

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    Area of Science:

    • Discrete Mathematics
    • Theoretical Computer Science
    • Graph Theory

    Background:

    • Boolean functions are fundamental in computer science and logic.
    • Graph-theoretic approaches offer novel perspectives for analyzing complex functions.
    • Understanding linearly separable functions is crucial for various computational tasks.

    Purpose of the Study:

    • To develop a novel graph-theoretic approach for studying Boolean functions.
    • To characterize linearly separable Boolean functions using graph structures.
    • To introduce and define a new class of graphs, termed hyperstars.

    Main Methods:

    • Transforming a Boolean function with n variables into an induced subgraph (Hf) of the n-dimensional hypercube.
    • Analyzing the structural properties of Hf to understand function characteristics.
    • Defining and investigating the properties of hyperstar graphs.

    Main Results:

    • The induced subgraph (Hf) of any linearly separable Boolean function is a hyperstar.
    • Demonstrated that hyperstar graphs possess fundamental properties related to linearly separable Boolean functions.
    • Established a direct link between Boolean function properties and hyperstar graph structures.

    Conclusions:

    • The proposed hyperstar graph structure provides a powerful tool for analyzing linearly separable Boolean functions.
    • This graph-theoretic perspective reveals new insights into the fundamental properties of these functions.
    • The novel approach offers a new framework for future research in Boolean function analysis.