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September 28, 2025

Characterization of graphs whose Laplacian eigenvalue 1 has multiplicity 2 c ( G ) + p ( G ) − 1

Key Points

Graphs are characterized where the multiplicity of eigenvalue 1 equals the expression related to cyclomatic number and pendant vertices.
Wen et al. (2022) established a foundational inequality linking multiplicity to graph structure for connected graphs not being cycles.
The Laplacian matrix is central to evaluating eigenvalue multiplicity, impacting understanding of graph properties.
This characterization enhances insights into connected graphs, emphasizing the relation between vertex degrees and eigenvalue multiplicity.

Abstract

Let G be a connected graph with vertex set V(G) and edge set E(G). The Laplacian matrix of G is defined as L(G)=D(G)−A(G), where D(G) is a diagonal matrix of degrees of the vertices of G and A(G) is the adjacency matrix of G. The multiplicity of an eigenvalue μ of L(G) is denoted by mG(μ). In 2022, Wen et al. Czech. Math. J., 72(2022) proved that, if G is not a cycle, then mG(μ)≤2c(G)+p(G)−1, where c(G)=|E(G)|−|V(G)|+1 is the cyclomatic number of G and p(G) is the number of pendant vertices of G. We characterize the graph G with mG(1)=2c(G)+p(G)−1.

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Characterization of graphs whose Laplacian eigenvalue 1 has multiplicity 2 c ( G ) + p ( G ) − 1

Key Points

Abstract

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