Double Corona Product Graphs

• Double corona product graphs are defined by applying two successive corona operations or twin-host embeddings, resulting in hierarchical, multi-layered network structures.
• They exhibit distinctive spectral, metric, and centrality properties that aid in analyzing equitable coloring, metric dimensions, and threshold processes.
• These constructions are applicable in fields such as network science and epidemiology, modeling multi-community systems with enhanced recursive and dynamic properties.

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 (Furmañczyk et al., 2012, Yero et al., 2010, Moon et al., 29 Aug 2025, Sharma et al., 2015).

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 $H$0, $H$1.
• The double corona is the case $H$2: $H$3.

The vertex set and edge set of $H$4 are: $H$5 Each “depth” in the hierarchy corresponds to a corona iteration; the process can be generalized for $H$6 (Furmañczyk et al., 2012, Yero et al., 2010, Sharma et al., 2015).

(b) Twin-Host Block Model (“GH”)

For certain irreversible threshold spread and block-propagation models, the double-corona product $H$7 is defined as follows (Moon et al., 29 Aug 2025):

• Take two disjoint isomorphic copies of $H$8 ($H$9), with vertex sets $G$0 and $G$1.
• Attach $G$2 disjoint copies of $G$3, denoted $G$4, each joined to the $G$5-th twin-vertex $G$6 and $G$7.
• The vertex set is $G$8.
• The edge set is $G$9.

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 $H$0:

• Total vertices: $H$1
• For the twin model $H$2: $H$3

Degree Distribution

In $H$4:

• An original vertex $H$5 has $H$6.
• Vertices at increasing “depth” gain $H$7 neighbors per corona iteration (Furmañczyk et al., 2012, Sharma et al., 2015).

For the twin model $H$8:

• $H$9
• For $|V(G)|=n$0, $|V(G)|=n$1 (Moon et al., 29 Aug 2025).

Diameter and Clustering

Each corona step increases diameter by $|V(G)|=n$2. If $|V(G)|=n$3, then $|V(G)|=n$4 (Sharma et al., 2015). Clustering spectra are preserved through iterations: if $|V(G)|=n$5 has global clustering $|V(G)|=n$6, then so does $|V(G)|=n$7.

3. Structural Properties and Centrality

Degree and Centrality Distributions

For regular seed graphs, the cumulative degree distribution of $|V(G)|=n$8 decays exponentially (Sharma et al., 2015). Specifically, the degree sequence splits into three classes:

• $|V(G)|=n$9, multiplicity $|V(H)|=m$0
• $|V(H)|=m$1, multiplicity $|V(H)|=m$2
• $|V(H)|=m$3, multiplicity $|V(H)|=m$4

For $|V(H)|=m$5, cumulative betweenness distribution in $|V(H)|=m$6 is asymptotically a power law with exponent $|V(H)|=m$7 (Sharma et al., 2015).

Connectivity

If $|V(H)|=m$8 is connected and $|V(H)|=m$9, $G \circ H$0 is connected for every $G \circ H$1 (Furmañczyk et al., 2012).

4. Spectral Properties

For $G \circ H$2 $G \circ H$3-regular:

(a) Adjacency Spectrum

For each $G \circ H$4 (an eigenvalue of $G \circ H$5): $G \circ H$6 with “step-2” eigenvalues: $G \circ H$7 each of multiplicity 1. The old eigenvalues are repeated with multiplicity $G \circ H$8 (Sharma et al., 2015).

(b) Laplacian Spectrum

For Laplacian eigenvalues $G \circ H$9: $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$0 with corresponding eigenvalues: $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$1 The algebraic connectivity is $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$2 and satisfies $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$3 (Sharma et al., 2015).

(c) Signless Laplacian Spectrum

For signless Laplacian eigenvalues $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$4: $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$5 producing eigenvalues: $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$6 with other $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$7 levels repeated (Sharma et al., 2015).

5. Metric Dimension

Let $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$8 and $V(G) \cup \{(v,h)\mid v\in V(G),\, h\in V(H)\}$9 be connected graphs with $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$0, $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$1:

Bounds and Key Results

• Lower bound: $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$2.
• If $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$3, then $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$4.
• General bound: $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$5.
• If $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$6 and either $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$7 or $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$8 is a large cycle, then $$E(G) \cup \{(v,h)(v,h') \mid hh' \in E(H)\} \cup \{(v,h)v \mid v\in V(G),\, h\in V(H)\}$$9 (Yero et al., 2010).

Resolving Sets

Any metric basis for $l$0 in each copy lifts to a metric basis for the double corona. For diameter $l$1, the union of metric bases of each $l$2-copy suffices.

Example

For $l$3 and $l$4: $l$5 as confirmed by explicit construction (Yero et al., 2010).

6. Coloring and Equitable Chromatic Number

Equitable coloring is recursively propagated in the corona product. Key results for double corona ($l$6):

General Bounds

• If $l$7 ($l$8), then $l$9 for all $H$00.
• For $H$01 $H$02-partite ($H$03), and $H$04 equitably $H$05-colorable with $H$06:

$H$07

• For $H$08 (even cycles), dichotomy:
  • If $H$09 is equitably 3-colorable and $H$10 or $H$11, then $H$12;
  • Otherwise, $H$13.
• For $H$14 (odd cycles), $H$15.
• For $H$16 (paths), $H$17 or 4, depending on $H$18 and divisibility conditions (Furmañczyk et al., 2012).

All constructive coloring algorithms are polynomial in the size of the double corona, provided an initial equitable $H$19-coloring of $H$20 is given.

Equitable Coloring Conjecture (ECC)

Every value of $H$21 proven above satisfies $H$22, confirming ECC for double coronas in all cases considered (Furmañczyk et al., 2012).

7. Saturation Dynamics and Threshold Processes

The irreversible $H$23-threshold process, where a vertex is colored upon having at least $H$24 colored neighbors, exhibits distinct thresholds on double corona products (Moon et al., 29 Aug 2025).

For $H$25, $H$26 (cycle and complete block):

$H$27

where $H$28 is the irreversible $H$29-threshold conversion number. The recursive formula: $H$30 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 (Moon et al., 29 Aug 2025).

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 (Moon et al., 29 Aug 2025).
• Social Influence: Modeling individuals occupying two social roles (eg, work/family), with attached community cliques (Moon et al., 29 Aug 2025).
• Network Science: Explaining block redundancy, resilience, and the effect of higher-order corona steps on connectivity, diameter, and clustering (Sharma et al., 2015).

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.

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Key References:

• (Furmañczyk et al., 2012) Equitable colorings: corona multi-products
• (Yero et al., 2010) Metric dimension: iterated coronas
• (Moon et al., 29 Aug 2025) Threshold conversion: twin-block coronas
• (Sharma et al., 2015) Spectral/structural properties: iterated corona graphs