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October 16, 2025Open Access

Spectral extrema of graphs forbidding a fan

Key Points

Graphs maximizing spectral radius without a fan subgraph are fully characterized for large n, suggesting a systematic approach.
The study shows that for graphs avoiding the fan structure H_(2k+2), there are also maximal graphs avoiding H_(2k+3) for large n.
By exploring spectral properties, this research provides insights into graph containment relationships based on subgraph restrictions.
Significant relationships among families of extremal graphs reveal underlying principles of graph structure and eigenvalue interdependencies.

Abstract

For a graph G, its spectral radius is the largest eigenvalue of its adjacency matrix. A fan H_ is a graph obtained by connecting a single vertex to all vertices of a path of order 4. Let SPEX (n, H_{) } be the set of all extremal graphs G of order n with the maximum spectral radius, where G contains no H_ as a subgraph. In this paper, we completely characterized the graphs in SPEX (n, H_{) } for any 4 and sufficiently large n. An interesting phenomenon was revealed: SPEX (n, H₂₊+₂) SPEX (n, H₂₊+₃) for any k1 and sufficiently large n.

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Cite This Study

Wenqian Zhang (Fri,) studied this question.

synapsesocial.com/papers/68f0f51d8dd8ea469b1d7179 https://doi.org/https://doi.org/10.48550/arxiv.2508.05911

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