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August 17, 2025Open Access

Eigenvalues and component factors in graphs

Key Points

The paper reveals conditions for graphs to contain a {k1,1,k1,2,...,k1,k,k1,2k+1}-factor based on eigenvalues.
Using spectral methods, tight conditions are established regarding spectral radius and distance spectral radius.
Analysis focuses on connected graphs, drawing from previous characterizations by Kano, Lu, and Yu.
Findings may enhance understanding of factor structures within graph theory and applications in network analysis.

Abstract

For a set H of connected graphs, an H-factor of G is a spanning subgraph F of G if each component of F is isomorphic to an element of H. Kano, Lu and Yu Electron. J. Combin. 26 (2019) P4. 33 provided a good characterization based on an isolated vertex condition for the existence of a \K₁, ₁, K₁, ₂, , K₁, ₊, T (2k+1) \-factor in graphs. Motivated by the above elegant result, we in this paper focus on the existence of a \K₁, ₁, K₁, ₂, , K₁, ₊, T (2k+1) \-factor in graphs from perspective of eigenvalues. By adopting a crucial technique due to Tait J. Combin. Theory Ser. A 166 (2019) 42-58 and combining typical spectral methods and structural analysis, we present tight sufficient conditions in terms of the spectral radius and the distance spectral radius for a graph to contain a \K₁, ₁, K₁, ₂, , K₁, ₊, T (2k+1) \-factor, respectively.

Eigenvalues and component factors in graphs

Key Points

Abstract

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