The minimum spectral radius of $t{P}_4$-saturated graphs

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 $t{P}_4$, proving that every $n$-vertex $t{P}_4$-saturated graph $G$ with $t\ge 2$ and $n\ge 4t$ satisfies $\ddot{I}(G)\ge \frac{1+\sqrt{17}}{2}$, and characterizing all $t{P}_4$-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 $t{P}_4$-saturated graphs minimizing the spectral radius is disjoint from that minimizing the number of edges.