October 1, 2025Open Access

Laplacian eigenvalue distribution and girth of graphs

Key Points

Graphs with girth greater than a specific threshold have limited Laplacian eigenvalues.
The number of Laplacian eigenvalues in a defined interval can be expressed in terms of the graph's girth.
Particular graph structures like cycles or specific configurations yield exact eigenvalue counts.
Findings illustrate the relationship between graph structure and eigenvalue distribution.

Abstract

Let G be a connected graph on n vertices with girth g. Let mGI denote the number of Laplacian eigenvalues of graph G in an interval I. In this paper, we show that if G is not a cycle, then mG (n-g+3, n] n-g. Moreover, we prove that mG (n-g+3, n]= n-g if and only if G C₃ or G K₃, ₂ or G U₁, where U₁ is obtained from a cycle by joining a single vertex with a vertex of this cycle.

Laplacian eigenvalue distribution and girth of graphs

Key Points

Abstract

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Also Consider

Also Consider