Minimum spectral radius of graphs of fixed order and dissociation number and its connection to Turán problems

TL;DR

This work investigates the minimum size and minimum spectral radius of connected graphs with fixed order $n$ and dissociation number $\tau$, linking these extremal problems to Turán-type questions for complete multipartite graphs. By introducing and exploiting forbidden substructures such as odd cocktail party graphs $L_{2k+1}$ and cocktail party graphs $CP_d$, the authors derive edge minimizers and, for several cases, identify spectral minimizers within $\mathcal{D}_{n,\tau}$. They generalize Erdős–Simonovits-type Turán results to broader families $K_{q+1}(r_1,\dots,r_{q+1})$ and establish stability results that connect near-extremal graphs to multipartite joins, including connected-complement variants. The paper also develops a cohesive framework relating $d$-independence number problems to Turán-type bounds, yielding asymptotic bounds on edges and spectral radii and furnishing explicit constructions (e.g., $\mathcal{T}_{n,2k}$ and aligned CP-paths) that achieve extremality. Collectively, these results illuminate the interplay between dissociation constraints and Turán-type extremal graphs, with implications for spectral graph theory and network design.

Abstract

Let $\mathcal{D}_{n,τ}$ be the set of all simple connected graphs of order $n$ and dissociation number $τ.$ In this paper, we study the minimum size and the minimum spectral radius of graphs in $\mathcal{D}_{n,τ}$ in connection with Turán-type problems for complete multipartite graphs. We characterize the Tur\' an graphs for several complete multipartite graphs where the size of one of the partite sets is much smaller than the size of the remaining partites. This extends a result of Erdős and Simonovits [16]. Additionally, we prove some stability results to get the structure of graphs without such a forbidden complete multipartite subgraph, and close to Turán number of edges. As an application, we show that a graph with the minimum spectral radius in $\mathcal{D}_{n,τ}$ must be a graph with the minimum size in $\mathcal{D}_{n, τ}$ when $n$ is sufficiently large and satisfies some parity conditions. We then describe a few structural properties of graphs with the minimum spectral radius in $\mathcal{D}_{n,τ}$. For even dissociation numbers and any order $n$, we compute the minimum size of a graph in $\mathcal{D}_{n,τ}$ and use it to characterize the graphs in $\mathcal{D}_{n, 4}$ that attain the minimum size and the minimum spectral radius. We also apply the stability results to upper bound the minimum number of edges and spectral radius for connected graphs with a given $d$-independence number when the order of the graph is sufficiently large. Finally, we derive two new bounds on the value of $τ(G)$ for a given graph $G$.

Figures (10)

• Figure 1: The graph in $G_{n,2k}$ when $n\equiv 0 \pmod {2k}.$
• Figure 2: The graphs in $G_{6,4}$.
• Figure 3: The graphs in $\mathcal{T}_{6,4}$.
• Figure 4: Graphs in $G^-_{5,4}.$
• Figure 5: Graphs on $6$ vertices with $10$ edges that are not in the set $G^-_{6,4}$.

Theorems & Definitions (99)

• Definition 1.1
• Definition 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.3
• proof
• Definition 3.1
• Lemma 3.2
• proof