The minimum spectral radius of $tP_4$-saturated graphs

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.

💡 Research Summary

The paper investigates the spectral saturation problem for the linear forest tP₄, i.e., the disjoint union of t copies of a four‑vertex path. A graph G is called F‑saturated if it contains no copy of F but the addition of any missing edge creates at least one copy of F. The authors focus on determining the smallest possible spectral radius ρ(G) among all n‑vertex tP₄‑saturated graphs and on characterizing the extremal graphs that attain this minimum.

The main result (Theorem 1.2) states that for any integer t ≥ 2 and any n‑vertex tP₄‑saturated graph G with n ≥ 4t, the spectral radius satisfies
 ρ(G) ≥ ρ(N₄) = (1 + √17)/2.
Equality holds if and only if G is the disjoint union of (t − 1) copies of a specific four‑vertex graph N₄ (illustrated in the paper) together with a residual component Z. The component Z belongs to a family 𝔖 defined as
 𝔖 = { ⊔{i=2}^{3} x_i K_i ⊔ ⊔{i=4}^{7} x_i K_{1,i−1} : x_i ≥ 0, ∑{i=2}^{7} i·x_i = n − 12t + 12 }.
In other words, Z is a multiset of complete graphs K₂, K₃, K₄, K₅, K₆, K₇ and stars K{1,1}, K_{1,2}, …, K_{1,6}, whose total order equals n − 12t + 12. This description is exact: any tP₄‑saturated graph attaining the lower bound must have precisely this structure, and conversely every graph of this form indeed reaches the bound.

The proof of the lower bound relies on a novel sufficient condition (Theorem 1.3). For a connected graph G, define for each vertex v
 F(v) = d(v)² + ∑{w∈N(v)∪N₂(v)} |N(w)∩N(v)|·d(w) − ∑{w∈N(v)} d(w)² − 3d(v) + 4.
If F(v) ≥ 0 for all vertices, then ρ(G) ≥ ρ(N₄); equality holds exactly when F(v)=0 for every v. The authors obtain this condition by applying Lemma 2.4 to the cubic polynomial p(x)=x³−2x²−3x+4, noting that p(ρ) ≥ min_v F(v). Since the roots of p(x) are (1+√17)/2, 1, and a negative number, any graph with p(ρ) ≥ 0 must satisfy ρ ≥ (1+√17)/2 (the negative root is irrelevant because a non‑trivial graph has ρ > 1). The equality case forces all row sums of p(A) to be equal, which translates precisely into the zero‑condition for F(v). This algebraic approach is complemented by a series of structural lemmas (Lemmas 2.5–2.7) that restrict the possible degree patterns in a tP₄‑saturated graph: vertices of degree 2 must belong to a triangle, a vertex of degree 1 cannot be adjacent to a vertex of degree 3, and various configurations involving degree‑3 vertices are ruled out. Lemma 2.2 (subdivision of an internal edge does not increase the spectral radius) is used repeatedly to argue that any extremal graph cannot contain long internal paths, pushing the structure toward the highly regular form described above.

Beyond the spectral bound, the authors compare the set SAT_sp(n, tP₄) of graphs minimizing the spectral radius with the classical edge‑minimal set SAT(n, tP₄). They prove that for t ≥ 3 and n ≥ 6t + 4, these two families are disjoint. This contrasts with many earlier results where the spectral extremal family is a subset of the edge‑extremal family (e.g., for K_{r+1}‑free graphs). The paper therefore demonstrates that, in the saturation context, minimizing the spectral radius can lead to completely different extremal configurations than minimizing the number of edges.

The paper is organized as follows. Section 2 collects definitions, notation, and known results on tP₄‑saturated graphs, and establishes several new lemmas about degree constraints and internal paths. Section 3 proves Theorem 1.3, developing the polynomial‑row‑sum inequality and analyzing the function F(v). Section 4 builds on this foundation to prove Theorem 1.2, handling various cases according to the degree of a vertex with degree 3 and employing the previously derived lemmas to eliminate all configurations that would force a larger spectral radius. The final part discusses the divergence between spectral and edge minimization for large t and n.

In summary, the authors provide a sharp lower bound for the spectral radius of tP₄‑saturated graphs, give a complete structural characterization of the extremal graphs, and reveal a fundamental difference between spectral and edge extremality in the saturation setting. Their method blends algebraic eigenvalue techniques with delicate combinatorial analysis, opening a pathway for future investigations of spectral saturation problems for other families of forbidden subgraphs.