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The minimum spectral radius of $tP_4$-saturated graphs

Junxue Zhang, Liwen Zhang

TL;DR

This work addresses the spectral-saturation problem for the forest $tP_4$ by establishing a universal lower bound $\rho(G) \ge \frac{1+\sqrt{17}}{2}$ for all $n$-vertex graphs that are $tP_4$-saturated with $t\ge 2$ and $n\ge 4t$, and provides a complete characterization of equality cases. The authors derive a local-structure condition via a cubic polynomial bound and a carefully defined function $F(v)$ that ties vertex neighborhoods to the spectral radius, enabling a sharp bound when $F(v)\ge 0$ for all $v$. They further prove that the minimum spectral radius is achieved precisely by graphs of the form $(t-1)N_4 \cup Z$, with $Z$ restricted to a forest built from $K_2$, $K_3$, and stars $K_{1,i-1}$ for $i=4..7$, subject to a linear constraint on vertex counts. Additionally, for certain parameter ranges (e.g., $t=2$, odd $n\ge 13$ or $t\ge 3$, $n\ge 6t+4$), the spectral-saturation minimizers are shown to be disjoint from edge-minimizers, highlighting a separation between spectral and edge considerations in saturation problems.

Abstract

A graph $G$ is called {\em$F$-saturated} if $G$ does not contain $F$ as a subgraph but adding any missing edge to $G$ creates a copy of $F$. In this paper, we consider the spectral saturation problem for the linear forest $tP_4$, proving that every $n$-vertex $tP_4$-saturated graph $G$ with $t\geq 2$ and $n\ge 4t$ satisfies $ρ(G)\ge \frac{1+\sqrt{17}}{2}$, and characterizing all $tP_4$-saturated graphs for which equality holds. Moreover, we obtain that, for $t=2$ with odd $n\ge 13 $, and for $t\ge 3$ with $n\ge 6t+4$, the set of $n$-vertex $tP_4$-saturated graphs minimizing the spectral radius is disjoint from that minimizing the number of edges.

The minimum spectral radius of $tP_4$-saturated graphs

TL;DR

This work addresses the spectral-saturation problem for the forest by establishing a universal lower bound for all -vertex graphs that are -saturated with and , and provides a complete characterization of equality cases. The authors derive a local-structure condition via a cubic polynomial bound and a carefully defined function that ties vertex neighborhoods to the spectral radius, enabling a sharp bound when for all . They further prove that the minimum spectral radius is achieved precisely by graphs of the form , with restricted to a forest built from , , and stars for , subject to a linear constraint on vertex counts. Additionally, for certain parameter ranges (e.g., , odd or , ), the spectral-saturation minimizers are shown to be disjoint from edge-minimizers, highlighting a separation between spectral and edge considerations in saturation problems.

Abstract

A graph is called {\em-saturated} if does not contain as a subgraph but adding any missing edge to creates a copy of . In this paper, we consider the spectral saturation problem for the linear forest , proving that every -vertex -saturated graph with and satisfies , and characterizing all -saturated graphs for which equality holds. Moreover, we obtain that, for with odd , and for with , the set of -vertex -saturated graphs minimizing the spectral radius is disjoint from that minimizing the number of edges.
Paper Structure (4 sections, 11 theorems, 21 equations, 2 figures)

This paper contains 4 sections, 11 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proof of Theorem \ref{['tp4suffcondi']}
  4. The Proof of Theorem \ref{['tp4spec']}

Key Result

Theorem 1.2

Let $t\ge 2$ and $G$ be a $tP_4$-saturated graph on $n$ vertices with $n\ge 4t$. Then with equality if and only if $G\cong (t-1)N_4\cup Z$, where $Z\in \{\bigcup_{i=2}^{3}x_iK_i\cup (\bigcup_{i=4}^{7} x_iK_{1,i-1}): x_i\ge 0$ and $\sum_{i=2}^{7} i x_i=n-12t+12\}$.

Figures (2)

  • Figure 1: The graphs $N_4$ and $N_4^*$
  • Figure 2: The graphs with $\rho(G)\geq \frac{1+\sqrt{17}}{2}$

Theorems & Definitions (13)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 3 more