The minimum spectral radius of $tP_4$-saturated graphs

TL;DR

This paper investigates the minimum spectral radius of graphs that are saturated with respect to the linear forest $tP_4$, establishing bounds and characterizations for such graphs and their extremal properties.

Contribution

It provides a lower bound on the spectral radius for $tP_4$-saturated graphs and characterizes the extremal graphs achieving equality.

Findings

Spectral radius of $tP_4$-saturated graphs is at least (1+√17)/2.

Characterization of graphs where the minimum spectral radius is attained.

The extremal graphs for spectral radius differ from those minimizing edges for certain parameters.

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 $\rho(G)\ge \frac{1+\sqrt{17}}{2}$, 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.