March 3, 2026

The Plans Theorem for Cones over Generalized I -Graphs

Key Points

The critical group of cones over generalized I-graphs demonstrates a direct relationship with the abelian group structure.
Key evidence shows that the structure is akin to the homology group behavior in topological spaces, enhancing our understanding of these graphs.
Utilizing the Plans theorem as a foundation, the approach explores the homological characteristics of related graph constructs.
Implications highlight the potential for new insights into graph theory and algebraic topology relationships, expanding the scope of existing models.

Abstract

We consider a family of graphs generalizing the family of I -graphs, which in turn includes generalized Petersen graphs and prismatic graphs. The paper is devoted to the study of the critical group of a graph that is a cone over a generalized I -graph. The main result of the article is an analog of the Plans theorem (1953), which describes the first homology group of an n -sheeted cyclic cover of the three-dimensional sphere branched over a knot. It asserts that this homology group is almost a direct sum of two copies of a certain abelian group. In this paper, analogous results are established for the structure of the critical group of the graphs under consideration.

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The Plans Theorem for Cones over Generalized I -Graphs

Key Points

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