A fundamental theorem on graph operators
A graph operator is a function [Formula: see text] defined on some set of graphs such that whenever two graphs G and H are isomorphic, written [Formula: see text], then [Formula: see text]. For a graph G not in the domain of [Formula: see text], we put [Formula: see text]. Also, let us define [Formu...
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| Format: | Article |
| Language: | English |
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World Scientific Publishing
2025-01-01
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| Series: | Mathematics Open |
| Subjects: | |
| Online Access: | https://www.worldscientific.com/doi/10.1142/S2811007225500105 |
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| Summary: | A graph operator is a function [Formula: see text] defined on some set of graphs such that whenever two graphs G and H are isomorphic, written [Formula: see text], then [Formula: see text]. For a graph G not in the domain of [Formula: see text], we put [Formula: see text]. Also, let us define [Formula: see text], and for any integer [Formula: see text], [Formula: see text] We prove that if [Formula: see text] is a graph operator, then the sequence [Formula: see text] has only three possible types of behavior. Either [Formula: see text] for some integer [Formula: see text], or [Formula: see text], or there exist integers [Formula: see text], [Formula: see text] such that the graphs [Formula: see text] are non-isomorphic ([Formula: see text], and [Formula: see text] for all integers [Formula: see text]. We illustrate this using two new graph operators, namely, the path graph operator and the claw graph operator. |
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| ISSN: | 2811-0072 |