---
title: Double Corona Product Graphs
url: https://www.emergentmind.com/topics/double-corona-product-graphs
type: topic
---

# Double Corona Product Graphs

A double corona product graph is a graph-theoretic construct produced from two graphs, $G$ and $H$, via a two-stage iterative corona product, or via a doubled host-block “twin” embedding, both of which are central in the analysis of metric, structural, spectral, and saturation dynamics of complex graphs. The construction serves as a foundational object in the study of equitable colorings, metric dimensions, threshold processes, and spectral characteristics, and is referenced under several notations in the literature depending on context and analytic focus [1210.6568, 1009.2586, 2508.21764, 1507.05335].

## 1. Formal Definitions and Construction

There are two principal constructions referred to as the double corona product in recent literature.

### (a) Iterated Corona Product

Given finite simple graphs $G$ and $H$ with $|V(G)|=n$ and $|V(H)|=m$:

- The **corona product** $G \circ H$ has vertex set $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$, and edge set
  $E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$.
- The **$l$-fold iterated corona product** is recursively defined as $G \circ^l H = (G \circ^{l-1} H) \circ H$, $l\ge2$.
- The **double corona** is the case $l=2$: $G \circ^2 H = (G \circ H) \circ H$.

The vertex set and edge set of $G \circ^2 H$ are:
\[
\begin{aligned}
V(G \circ^2 H) &= V(G) \cup \{(v,i) : v\in V(G), i\in V(H)\} \cup \{((v,i), j) : v\in V(G), i,j\in V(H)\}, \\
E(G \circ^2 H) &= E(G) \cup \{(v,i)(v,i') : i i' \in E(H)\} \cup \{(v,i)v \} \\
&\quad \cup \{((v,i), j)((v,i),j') : j j' \in E(H)\} \cup \{((v,i),j)(v,i)\}.
\end{aligned}
\]
Each “depth” in the hierarchy corresponds to a corona iteration; the process can be generalized for $l>2$ [1210.6568, 1009.2586, 1507.05335].

### (b) Twin-Host Block Model (“GH”)

For certain irreversible threshold spread and block-propagation models, the **double-corona product** $GH$ is defined as follows [2508.21764]:

- Take two disjoint isomorphic copies of $G$ ($G', G''$), with vertex sets $\{v_1',...,v_n'\}$ and $\{v_1'',...,v_n''\}$.
- Attach $n$ disjoint copies of $H$, denoted $H^i$, each joined to the $i$-th twin-vertex $v_i'$ and $v_i''$.
- The vertex set is $V(GH) = V(G') \cup V(G'') \cup_{i=1}^n V(H^i)$.
- The edge set is $E(GH) = E(G') \cup E(G'') \cup_{i=1}^n E(H^i) \cup \cup_{i=1}^n \{ (v_i',x), (v_i'',x) : x\in V(H^i) \}$.

Both constructions produce graphs with complex, highly recursive community structures, distinct degree sequences, and potential for nontrivial metric, dynamic, and spectral properties.

## 2. Basic Graph-Theoretic Parameters

### Vertex and Edge Counts

For the iterated double-corona $G \circ^2 H$:
- Total vertices: $|V(G)| \cdot (m+1)^2$
- For the twin model $GH$: $2n + n m = n(2 + m)$

### Degree Distribution

In $G \circ^2 H$:
- An original vertex $v \in V(G)$ has $\deg_{G \circ^2 H}(v) = \deg_G(v) + 2m$.
- Vertices at increasing “depth” gain $m$ neighbors per corona iteration [1210.6568, 1507.05335].

For the twin model $GH$:
- $\deg_{GH}(v_i') = \deg_G(v_i) + m$
- For $w_j^i \in V(H^i)$, $\deg_{GH}(w_j^i) = \deg_H(w_j) + 2$ [2508.21764].

### Diameter and Clustering

Each corona step increases diameter by $+2$. If $\operatorname{diam}(G) = D^0$, then $\operatorname{diam}(G^{(2)}) = D^0 + 4$ [1507.05335]. Clustering spectra are preserved through iterations: if $G$ has global clustering $C^0$, then so does $G^{(2)}$.

## 3. Structural Properties and Centrality

### Degree and Centrality Distributions

For regular seed graphs, the cumulative degree distribution of $G^{(2)}$ decays exponentially [1507.05335]. Specifically, the degree sequence splits into three classes:
- $D_0(v) = d_G(v) + 2n$, multiplicity $n$
- $D_1(v) = d_G(v) + (n+1)$, multiplicity $n^2$
- $D_2(v) = d_G(v) + 1$, multiplicity $n^3$

For $G=K_n$, cumulative betweenness distribution in $G^{(2)}$ is asymptotically a power law with exponent $\approx2$ [1507.05335].

### Connectivity

If $G$ is connected and $|V(H)| \geq 1$, $G \circ^l H$ is connected for every $l \geq 1$ [1210.6568].

## 4. Spectral Properties

For $G$ $r$-regular:

### (a) Adjacency Spectrum

For each $\mu_i$ (an eigenvalue of $G$):
\[
z_1 = \sqrt{(r-\mu_i)^2 + 4n}, \quad
z_2 = \sqrt{(r-\mu_i + z_1)^2 + 16n},
\]
with “step-2” eigenvalues:
\[
\lambda_i^{\pm} = \frac{\mu_i + 3r \pm (z_1 + z_2)}{4},
\]
each of multiplicity 1. The old eigenvalues are repeated with multiplicity $n(n+1)$ [1507.05335].

### (b) Laplacian Spectrum

For Laplacian eigenvalues $\nu_i$:
\[
Z_1 = \sqrt{(\nu_i + n + 1)^2 - 4\nu_i}, \;\;
Z_2 = \sqrt{(\nu_i + 3(n+1) + Z_1)^2 - 8(\nu_i + n+1)},
\]
with corresponding eigenvalues:
\[
\Lambda_i^{\pm} = \frac{\nu_i + 3(n+1) \pm (Z_1 + Z_2)}{4}.
\]
The algebraic connectivity is $\min_{i>1}\{\Lambda_i^-\}$ and satisfies $\Lambda_2^- < 1$ [1507.05335].

### (c) Signless Laplacian Spectrum

For signless Laplacian eigenvalues $q_i$:
\[
w_1 = \sqrt{(q_i + n - (2r+1))^2 + 4n}, \quad
w_2 = \sqrt{(q_i + n + 2r +1 + w_1)^2 + 16n},
\]
producing eigenvalues:
\[
\Theta_i^{\pm} = \frac{q_i + 3n + 2r +1 \pm (w_1 + w_2)}{4},
\]
with other $q_i + 1$ levels repeated [1507.05335].

## 5. Metric Dimension

Let $G$ and $H$ be connected graphs with $|V(G)|=n_1$, $|V(H)|=n_2$:

### Bounds and Key Results

- Lower bound: $\dim(G \odot^2 H) \geq n_1 (n_2+1) \dim(H)$.
- If $\operatorname{diam}(H)\leq2$, then $\dim(G \odot^2 H) = n_1 (n_2+1) \dim(H)$.
- General bound: $\dim(G \odot^2 H) \leq n_1 (n_2+1) \dim(K_1 \odot H)$.
- If $n_2>7$ and either $\operatorname{diam}(H)>6$ or $H$ is a large cycle, then $\dim(G \odot^2 H) = n_1 (n_2+1) \dim(K_1 \odot H)$ [1009.2586].

### Resolving Sets

Any metric basis for $K_1 \odot H$ in each copy lifts to a metric basis for the double corona. For diameter $\leq2$, the union of metric bases of each $H$-copy suffices.

### Example

For $G = P_2$ and $H = K_3$:
\[
\dim(G \odot^2 H) = 2 \times 4 \times 2 = 16
\]
as confirmed by explicit construction [1009.2586].

## 6. Coloring and Equitable Chromatic Number

Equitable coloring is recursively propagated in the corona product. Key results for double corona ($l=2$):

### General Bounds

- If $\chi(G) \leq m+1$ ($H = K_m$), then $\chi_=(G \circ^l K_m) = m+1$ for all $l$.
- For $H$ $r$-partite ($r\leq k-1$), and $G$ equitably $k$-colorable with $k | |V(G)|$:
  \[
  \chi_=(G \circ^l H) = k.
  \]
- For $H = C_{2k}$ (even cycles), dichotomy:
    - If $G$ is equitably 3-colorable and $3 | n$ or $k=2$, then $\chi_=(G \circ^l C_{2k}) = 3$;
    - Otherwise, $\chi_=(G \circ^l C_{2k}) = 4$.
- For $H = C_{2k+1}$ (odd cycles), $\chi_=(G \circ^l C_{2k+1}) = 4$.
- For $H = P_m$ (paths), $\chi_=(G \circ^l P_m)=3$ or 4, depending on $m$ and divisibility conditions [1210.6568].

All constructive coloring algorithms are polynomial in the size of the double corona, provided an initial equitable $k$-coloring of $G$ is given.

### Equitable Coloring Conjecture (ECC)

Every value of $\chi_=(G \circ^2 H)$ proven above satisfies $\chi_=(G \circ^2 H) \leq \Delta(G \circ^2 H)$, confirming ECC for double coronas in all cases considered [1210.6568].

## 7. Saturation Dynamics and Threshold Processes

The irreversible $k$-threshold process, where a vertex is colored upon having at least $k$ colored neighbors, exhibits distinct thresholds on double corona products [2508.21764].

For $G=C_n$, $H=K_p$ (cycle and complete block):

\[
C_k(C_n K_p) = 
\begin{cases}
(k-2) n + \left\lfloor \frac{3n+3}{4} \right\rfloor, & k\leq p \\
pn, & k=p+1 \\
pn + 2\left\lceil \frac{n}{2} \right\rceil, & k=p+2 \\
\text{\(k\)-inconvertible}, & k\geq p+3
\end{cases}
\]

where $C_k(\cdot)$ is the irreversible $k$-threshold conversion number. The recursive formula:
\[
C_k(C_n K_p) = \tau n + C_{k-\tau}(C_n K_{p-\tau}),\quad \tau = \min\{k-2, p\}
\]
captures the forced seeding within each block at different thresholds. This quantifies how many initiators are required to trigger a global cascade in double-corona structures.

Probabilistic seeding analysis for double coronas remains open; deterministic bounds are sharp and reveal the amplification (or slowing) of saturation dynamics versus single corona products [2508.21764].

## 8. Applications and Implications

Double corona products encode hierarchical, multi-community structures with nontrivial centrality, threshold, and coloring properties.
- **Epidemiology:** Double-layer host–community clustering models; relevant to tracing and outbreak dynamics [2508.21764].
- **Social Influence:** Modeling individuals occupying two social roles (eg, work/family), with attached community cliques [2508.21764].
- **Network Science:** Explaining block redundancy, resilience, and the effect of higher-order corona steps on connectivity, diameter, and clustering [1507.05335].

A plausible implication is that saturated or equitable properties in single corona models often generalize or amplify in double coronas, with all main bounds remaining robust under strong structural or degree growth.

---

**Key References:**  
- [1210.6568] Equitable colorings: corona multi-products  
- [1009.2586] Metric dimension: iterated coronas  
- [2508.21764] Threshold conversion: twin-block coronas  
- [1507.05335] Spectral/structural properties: iterated corona graphs

Source: https://www.emergentmind.com/topics/double-corona-product-graphs