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Number of Components in the Graph Exponential with Complete Graph Exponents
Version 1
: Received: 24 May 2024 / Approved: 24 May 2024 / Online: 27 May 2024 (11:09:46 CEST)
How to cite: Holly, M. Number of Components in the Graph Exponential with Complete Graph Exponents. Preprints 2024, 2024051720. https://doi.org/10.20944/preprints202405.1720.v1 Holly, M. Number of Components in the Graph Exponential with Complete Graph Exponents. Preprints 2024, 2024051720. https://doi.org/10.20944/preprints202405.1720.v1
Abstract
Although Lov\'{a}sz introduced the graph exponential $G^K$ in 1967, this product has received little attention. This paper focuses on the number of components in $G^K$ when exponent $K$ is a complete graph $K_n$. For a connected bipartite $G$, when $K$ is $K_2$, $G^{K_2}$ has three components; and when $K$ is $K_n$ with $n>2$, $G^{K_n}$ produces $\lceil\frac{n+1}{2}\rceil$ components. A connected $G$ with an odd cycle produces one component in $G^{K_n}$.
Keywords
graph products; graph exponentiation; graph exponential; graph product components
Subject
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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