A characterization on trees $T$ with $m(T, λ)=p(T)-2$

Abstract

Let $m(G,λ)$ be the multiplicity of an eigenvalue $λ$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\neq C_n$, $m(G, λ) \leq 2c(G) + p(G) -1$, where $c (G) = |E(G)| - |V (G)| + 1$ and $p(G)$ are the cyclomatic number and the number of pendant vertices of $G$, respectively. In the same paper, they proposed the problem to characterize all connected graphs $G$ with eigenvalue $λ$ such that $m(G, λ) =2c (G)+ p(G)-1$. Wong et al. [Discrete Math. 347(2024), 113845] solved this problem for the case when $G$ is a tree by characterizing all trees $T$ with eigenvalue $λ$ such that $m(T , λ) = p(T )-1$. In this paper, we further provide the structural characterization on trees $T$ with eigenvalue $λ$ such that $m(T , λ) = p(T )-2$.

Figures (3)

• Figure 1: A flow chart on how to construct a tree $T$ in $\Gamma_{4}(0)$
• Figure 2: Some trees in $\Gamma^{2}_{4}(0)$
• Figure 3: Some trees in $\Gamma^{2}_{3}(2\cos \frac{\pi}{5})$

Theorems & Definitions (8)

• Theorem 1.1: Wang1
• Theorem 1.3: Wong
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Parter-Wiener Theorem, Johnson
• Lemma 2.2: Johnson
• Lemma 2.3: Wong
• Lemma 2.4: Johnson