Minimally k-Edge-Connected Graphs

• Minimally k-edge-connected graphs are defined by their property that the removal of any edge decreases connectivity from k to k-1, ensuring a tight structural configuration.
• They exhibit distinctive structural and spectral properties, often constructed with dominating vertices and regular subgraphs to achieve extremal frameworks.
• Efficient k-tree based algorithms enable the construction of these graphs, with applications in combinatorial theory, spectral analysis, and group-theoretic graph models.

A minimally $k$-edge-connected graph is a finite or infinite undirected simple graph $G = (V,E)$ with edge-connectivity $\lambda'(G)=k$, such that each edge $e\in E$ is critical for $k$-edge-connectivity: removing any $e$ yields a graph $G-e$ with $\lambda'(G-e)=k-1$. This property ensures that the global connectivity cannot be maintained upon deletion of any edge, signifying a structurally tight configuration where minimality, degree, regularity, and dominance phenomena interplay. These graphs are central in both extremal combinatorial theory and spectral graph theory, and underpin group-theoretic applications via power graphs.

1. Precise Definitions and Characterization

Let $G=(V,E)$ be a finite simple connected graph.

• Edge-connectivity $\lambda'(G)$: the minimal size of a set of edges whose removal disconnects $G$.
• Minimum degree $\delta(G) = \min\{\deg(v): v\in V\}$.
• $k$-Edge-connected: $\lambda'(G)=k$.
• Minimally $k$-edge-connected: For every $e\in E$, $\lambda'(G-e)=k-1$.

Theorem 2.1 from (Parveen et al., 2024) provides a full combinatorial characterization in the presence of a dominating vertex:

Let $G$ be a non-complete connected graph with a dominating vertex $x$ (i.e., $x$ adjacent to every other vertex). Then $G$ is minimally $k$-edge-connected for $k=\lambda'(G)$ if and only if:

1. $x$ is the unique dominating vertex.
2. $G-\{x\}$ is regular.

Explicitly: writing $\Delta = \deg(x)$ and $H = G - \{x\}$, $G$ is minimally $\Delta$-edge-connected if and only if $x$ is the only vertex with $\deg(x)=|V|-1$ and $H$ is $r$-regular with $r=\Delta-1$ (Parveen et al., 2024).

For arbitrary graphs (without a dominating vertex), all minimally $k$-edge-connected graphs have $\delta(G)=k$; no cycle carries a chord; and at least two vertices of degree $k$ must exist in the finite case (Stein, 2011).

2. Structural Properties and Extremal Examples

Structural extremality is governed by the interplay of degree sequences, regularity, dominance, and bipartition. For finite graphs (Stein, 2011), at least two vertices of degree $k$ are always present, and for $k \neq 1,3$, a linear fraction $c'_k \approx 1/2$ of the vertices have degree $k$.

The decomposition in (Parveen et al., 2024) underlies strong algebraic uniformity: for group-theoretic graphs, minimal edge-connectivity corresponds to regular substructures plus a dominating element, translating uniformity of group element orders into graph-theoretic minimality.

Extremal constructions (with $k \geq 2$):

• $k=2$: Let $H$ be a 1-regular graph (“perfect matching”) on $2m$ vertices and $x$ adjacent to all of $H$. $\delta(G)=2$, $G-\{x\}$ is 1-regular (Parveen et al., 2024).
• $k=3$: Let $H$ be a 2-regular graph (disjoint cycles) of $n$ vertices, with $x$ adjacent to all of $H$. $\delta(G)=3$, $G-\{x\}$ is 2-regular (Parveen et al., 2024).
• General: The complete bipartite graph $K_{k,n-k}$ achieves edge-count saturation and spectral extremality for large $k$ and $n$ (Lou et al., 13 Mar 2025).

For infinite graphs, similar degree phenomena hold only after incorporating “ends” (equivalence classes of rays). One always has at least two small points—vertices of degree $k$ or ends of edge-degree $k$—with new combinatorial phenomena at infinity (Stein, 2011).

3. Spectral Extremality and the Max–Min Problem

Spectral analogues of edge-extremal problems feature prominently. Given $G$ on $n$ vertices, the maximal spectral radius $\rho(G)$ among all minimally $k$-edge-connected graphs is achieved at the complete bipartite graph $K_{k,n-k}$ for $k \geq 3$ and large $n$ (Lou et al., 13 Mar 2025). In particular,

$\rho(G) \leq \rho(K_{k,n-k}) = \sqrt{k(n-k)}$

with equality only when $G \cong K_{k,n-k}$ (Lou et al., 13 Mar 2025). This also attains maximal edge-count $e(G)=k(n-k)$ for $n \geq 3k$ (Lou et al., 13 Mar 2025).

The $\alpha$-index $\varphi_\alpha(G)$, defined as the largest eigenvalue of $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for $\alpha \in [0,1]$, is maximized (for $k=2$) by the fan graph $F_{n-1} = K_1 \vee ((n-1)/2$ copies of $K_2)$ for odd $n$, and $K_{2,n-2}$ for even $n$ (Lou et al., 2023). Closed-form expressions for $\varphi_\alpha$ are given explicitly therein.

These results confirm that edge and spectral extremality coincide in the class of minimally $k$-edge-connected graphs of fixed order and connectivity.

4. Algorithmic Aspects and Constructions

Efficient algorithms exist for generating minimally $k$-edge-connected graphs from $k$-trees (chordal graphs of treewidth $k$):

• In a $k$-tree, edges with both endpoints of degree $\geq k+1$ are insensitive: their removal maintains $k$-edge-connectivity. Thus, iteratively deleting such edges yields a minimally $k$-edge-connected graph (Badarla et al., 2011).
• For $k=2$ (from a 2-tree), one enumerates triangles, deletes edges present in multiple triangles, and obtains a minimal structure—a “triangulated cycle” (Badarla et al., 2011).
• Complexity is $O(n^2)$, governed by triangle enumeration and edge-deletion steps.

This certifies the minimality property: no edge can be removed without reducing the edge-connectivity below $k$.

5. Average Edge-Connectivity and Bipartite Structure

Let $\lambda_G(u,v)$ denote the maximum number of edge-disjoint $u$–$v$ paths. The average edge-connectivity is

$$\overline{\lambda}(G) = \frac{1}{\binom{n}{2}} \sum_{\{u,v\}\subset V(G)} \lambda_G(u,v)$$

For optimal minimally $k$-edge-connected graphs (maximizing $\overline{\lambda}(G)$ on order $n$):

• Conjecture (Mol et al., 2021): For $k\geq 3$, the extremal graphs are bipartite—one part with all vertices of degree $k$, the other part with vertices of degree $>k$.
• Universal bound: For degree-partitioned minimally $k$-edge-connected graphs of order $n\geq2k+1$,

$$\overline{\lambda}(G) < k + \frac{k(n-2)^2}{8n(n-1)} < \frac{9k}{8}$$

Asymptotic constructions (family $\{I_{k,p}\}$) realize $\overline{\lambda}(I_{k,p}) \to 9k/8$ as $p\to\infty$ (Mol et al., 2021). Parallel statements hold for minimally $k$-connected graphs (vertex-connectivity).

6. Infinite Graphs, Ends, and Generalizations

In infinite graphs, minimality extends to “ends.” An end is an equivalence class of rays (one-way infinite paths) not separated by any finite vertex-set. Edge-degree of an end $\omega$, $d_e(\omega)$, is the maximum number of edge-disjoint rays in $\omega$. The main extension (Stein, 2011):

• Every (finite or infinite) edge-minimally $k$-edge-connected graph has at least two “small points”: vertices of degree $k$ or ends with $d_e(\omega) = k$.
• There exist infinite edge-minimally $k$-edge-connected graphs with no vertices of degree $k$, only ends with $d_e(\omega) = k$.

Open questions remain regarding the abundance of such small-degree vertices or ends in infinite settings, and whether every infinite minimally $k$-edge-connected graph contains infinitely many such points.

7. Applications to Group-Theoretic Graphs and Further Corollaries

The minimal edge-connectivity property underpins several algebraic graph constructions:

• Power graph $P(G)$ of a finite group $G$: minimally edge-connected (non-complete) exactly when $G$ is non-cyclic of prime exponent (Parveen et al., 2024).
• Enhanced power graph $PE(G)$: minimal edge-connectivity equivalent to all maximal cyclic subgroups having equal order and trivial intersections (in nilpotent $G$ this forces $G$ to be a $p$-group of exponent $p$) (Parveen et al., 2024).
• Order superpower graph $S(G)$: minimal edge-connectivity iff $G$ is a $p$-group (Parveen et al., 2024).

Further, in nilpotent groups, the minimal degree and vertex connectivity of $S(G)$ are equal precisely under minimal edge-connectivity (Parveen et al., 2024). The algebraic regularity and dominance properties in these graphs mirror the combinatorial minimality conditions.

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This overview provides the rigorous combinatorial, spectral, algorithmic, and algebraic landscape of minimally $k$-edge-connected graphs, synthesizing extremal results, structural decompositions, infinite generalizations, and their manifestations in algebraic graph theory.