A characterization of graphs $G$ with nullity $n(G)-d(G)-1$

Abstract

For a connected graph $G$ with order $n$, let $e(G)$ represent the number of its distinct eigenvalues, and let $d$ denote its diameter. We denote the eigenvalue multiplicity of $μ$ in $G$ by $m_G(μ)$. It is well established that the inequality $e(G) \geq d + 1$ implies that when $μ$ is an eigenvalue of $P_{d+1}$, it follows that $m_G(μ) \leq n - d$; otherwise, for any real number $μ$, we have $m_G(μ) \leq n - d - 1$. A graph is termed minimal if $e(G) = d + 1$. In 2013, Wong et al. characterized all minimal graphs for which $m_G(0) = n - d$. In this article, we provide a complete characterization of the graphs $G$ such that $m_G(0) = n - d - 1$.

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Theorems & Definitions (15)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7