Vertex-Corona in Graph Theory

• Vertex-corona is a graph operation that attaches a copy of one graph to each vertex of a base graph, altering distance, spectral, and coloring properties.
• The Laplacian formulation of vertex-corona uses a block matrix structure that facilitates exact calculations of resistance distances and eigenvalues.
• This construction supports many extensions—including variants for signed graphs, digraphs, and hypergraphs—enabling applications in network analysis, epidemiology, and combinatorial optimization.

Searching arXiv for recent and foundational papers on vertex-corona / corona products. arXiv search query: "vertex-corona corona product graphs resistance distance Laplacian generalized inverse" Vertex-corona is the graph operation more commonly called the corona product in the cited literature. For graphs $G$ and $H$, it is formed by taking one copy of $G$, taking one copy of $H$ for each vertex of $G$, and joining each vertex of $G$ to every vertex in its corresponding copy of $H$. This construction serves as a canonical attachment mechanism for studying how local grafting of repeated subgraphs alters distance, spectrum, coloring, domination, and propagation phenomena, and it has been extended to signed graphs, digraphs, hypergraphs, and several subdivision-based and neighborhood-based variants (Liu et al., 2015, Yero et al., 2012, Cavers et al., 17 Sep 2025, Kurian et al., 2024).

1. Definition and notation

Let $G_1$ and $G_2$ be graphs on disjoint vertex sets, with $|V(G_1)|=n_1$ and $H$0. The corona of $H$1 and $H$2, denoted in the cited literature by either $H$3 or $H$4, is obtained from one copy of $H$5 and $H$6 copies of $H$7 by joining the $H$8-th vertex of $H$9 to every vertex in the $G$0-th copy of $G$1 (Liu et al., 2015, Yero et al., 2012). In set-theoretic form,

$G$2

where $G$3 denotes the copy attached to $G$4 (Klavžar et al., 2020).

A recurrent source of confusion is not the construction itself but the notation. In the cited papers, the same vertex-attached product is written as $G$5 and $G$6, while other symbols are overloaded for different operations. For example, one paper uses $G$7 for neighborhood corona, whereas another uses $G$8 for edge corona (Liu et al., 2015, Klavžar et al., 2020). A second potential ambiguity is terminological: the “corona-vertex of the subdivision graph” is a distinct construction built from subdivision graphs, not the ordinary vertex-corona (Liu et al., 2016).

The elementary combinatorial effect of the operation is immediate in special families. For the cyclic base $G$9 and complete attachment $H$0, the graph $H$1 has $H$2 vertices, and the analogous double corona $H$3 has $H$4 vertices (Moon et al., 29 Aug 2025). This block structure underlies most later exact formulas.

2. Linear-algebraic formulation

The Laplacian description of the vertex-corona is particularly tractable when $H$5 is $H$6-regular and $H$7 is arbitrary. With a suitable labeling, the Laplacian of $H$8 has the block form

$H$9

where $G$0 is the all-1 vector and $G$1 is the Kronecker product (Liu et al., 2015). This decomposition makes the attachment mechanism explicit: the base-graph block is shifted by $G$2, while each attached copy contributes through a tensor-product term.

At the level of basic coloring, the ordinary chromatic number is controlled by the chromatic number of the factors: $G$3 The additional $G$4 reflects the fact that the central vertex of each corona block must avoid the colors used in its attached copy (Yero et al., 2012).

The operation also admits flexible generalizations in which the attachment is constrained to prescribed subsets. If $G$5 has vertices $G$6, $G$7, and each $G$8, then the generalized corona constrained by vertex subsets joins $G$9 only to the vertices of $G$0. Its adjacency characteristic polynomial is

$G$1

with

$G$2

where $G$3 is the coronal constrained by an index set (Rajkumar et al., 2020). This places the ordinary vertex-corona in a broader coronal-based framework.

3. Resistance distance and spectral theory

A major algebraic development for vertex-corona graphs is the use of Laplacian generalized inverses to compute effective resistances. For any $G$4-inverse $G$5 of a graph Laplacian,

$G$6

which is Klein and Randić’s formula for resistance distance (Liu et al., 2015). For $G$7, a symmetric $G$8-inverse is obtained from the block quantities

$G$9

and

$H$0

yielding

$H$1

as a symmetric $H$2-inverse of $H$3 (Liu et al., 2015). The explicit examples $H$4 and $H$5 in that work show how the full resistance-distance matrix can be extracted.

Resistance-based analysis extends beyond the ordinary vertex-corona. For the corona-vertex of the subdivision graph $H$6, the number of vertices is

$H$7

the number of edges is

$H$8

and the resistance formulas are expressed through $H$9, $G_1$0, and Kronecker products (Liu et al., 2016). The same paper also gives a Kirchhoff-index formula in terms of $G_1$1, $G_1$2, $G_1$3, the Laplacian eigenvalues of $G_1$4, and $G_1$5 (Liu et al., 2016). This suggests that the vertex-corona viewpoint is especially compatible with electrical-network invariants and resistance-based topological indices.

Spectral techniques have likewise been extended to multiple corona-derived operations. The subdivision-vertex neighbourhood corona and subdivision-edge neighbourhood corona admit closed adjacency, Laplacian, and signless Laplacian characteristic polynomials in terms of the spectra of the factors and matrix coronals, and these formulas were used to construct infinitely many pairs of cospectral graphs and new families of expander graphs (Liu et al., 2012).

4. Coloring, domination, and game parameters

The vertex-corona is a prolific testbed for refined coloring parameters. For distance-$G_1$6 coloring, the cited literature gives, among others,

$G_1$7

$G_1$8

and, for $G_1$9,

$G_2$0

There are also exact formulas for trees and cycles in distance-2 and distance-3 settings (Yero et al., 2012).

For domination-related colorings, the corona product exhibits a sharp separation between dominated and dominator chromatic behavior. The dominated chromatic number satisfies

$G_2$1

whereas the dominator chromatic number satisfies

$G_2$2

For example,

$G_2$3

These formulas show that the corona structure allows each central vertex of $G_2$4 to act as a singleton dominator class while reusing an optimal palette on the attached copies (Klavžar et al., 2020).

Classical domination parameters also admit exact corona formulas. If $G_2$5 has order $G_2$6 and $G_2$7 has order $G_2$8, then

$G_2$9

For $|V(G_1)|=n_1$0,

$|V(G_1)|=n_1$1

and the distance-$|V(G_1)|=n_1$2 domination number satisfies

$|V(G_1)|=n_1$3

Further exact formulas are given for independence domination, the domatic number, and the idomatic number (Yero et al., 2012).

The Maker-Breaker domination game on corona products is governed primarily by the second factor. For $|V(G_1)|=n_1$4, if Dominator can win on $|V(G_1)|=n_1$5 or if the first player can win on $|V(G_1)|=n_1$6, then

$|V(G_1)|=n_1$7

if Staller can win on $|V(G_1)|=n_1$8, then

$|V(G_1)|=n_1$9

When $H$00, the sharp exact formula

$H$01

holds (Divakaran et al., 2024). This makes the vertex-corona unusual among graph products: many game outcomes are inherited almost entirely from the attached factor.

5. Propagation processes and labeling invariants

Corona graphs are also a natural setting for irreversible threshold dynamics. For the irreversible $H$02-threshold process on $H$03, the irreversible $H$04-threshold conversion number is

$H$05

The associated random-seeding success probabilities are likewise given explicitly, and the paper defines the resilience factor as

$H$06

A parallel theory is developed there for double corona product graphs $H$07 (Moon et al., 29 Aug 2025). The stated applications are epidemiology and social influence modeling, with the block structure acting as a bottleneck for propagation.

Local antimagic labeling yields another exact parameter family. For $H$08 and $H$09,

$H$10

For cycles, the behavior splits by parity: for even $H$11 and any $H$12,

$H$13

whereas for $H$14,

$H$15

For complete graphs,

$H$16

and, in the one-leaf case,

$H$17

These formulas are derived by explicit edge labelings and lower-bound arguments based on leaf structure and vertex sums (Arumugam et al., 2018).

A correction to earlier lower-bound proofs was later supplied for friendship and fan graph coronas with null graphs. In particular, for $H$18,

$H$19

and the lower bounds

$H$20

were established with complete proofs (Lau et al., 2022).

6. Variants, extensions, and broader significance

The ordinary vertex-corona has generated a substantial family of related constructions. Among graph-theoretic variants are the neighborhood corona, edge corona, subdivision-vertex neighbourhood corona, $H$21-vertex corona, $H$22-vertex neighborhood corona, and the subdivision vertex-edge neighbourhood vertex-corona (Liu et al., 2015, Liu et al., 2012, K et al., 2024, Mukherjee et al., 15 Sep 2025, Wen et al., 2018). These variants preserve the central idea of attaching copies along vertex-indexed positions but alter the attachment interface from the vertex itself to its neighborhood, to inserted subdivision vertices, or to vertices arising from $H$23 or $H$24.

The operation also extends beyond ordinary graphs. For signed graphs, duplication add vertex corona product and duplication vertex corona product are switching isomorphic, and their adjacency, Laplacian, and signless Laplacian spectra can be written in terms of the spectra and coronals of the factors (Sonar et al., 2023). For digraphs, the symmetric-vertex-corona $H$25 has adjacency matrix

$H$26

and its adjacency, Laplacian, and signless Laplacian characteristic polynomials are expressed through digraph coronals (Cavers et al., 17 Sep 2025). For hypergraphs, the corona $H$27 generalizes the graph vertex-corona, and its adjacency and Seidel spectra are determined when $H$28 is regular (Kurian et al., 2024).

The locating-chromatic number has also been analyzed in the ordinary corona setting. If $H$29 is connected and $H$30 has components $H$31, then

$H$32

with sharpness examples for both bounds (Syofyan et al., 2024). In combinatorial commutative algebra, multi-clique corona graphs generalize clique-corona graphs and multi-whisker graphs; they are vertex decomposable and hence sequentially Cohen–Macaulay, with

$H$33

and

$H$34

The vertex-corona appears there as the case where all attached graphs are $H$35 (Muta et al., 28 Nov 2025).

Across these lines of work, the vertex-corona functions as a controlled attachment operator whose repeated-copy structure is simultaneously combinatorial and algebraic. The cited applications include chemical and physical index computation, genetic interaction networks, epidemiology, social influence modeling, cospectral graph construction, expander construction, and the study of edge ideals (Liu et al., 2015, Klavžar et al., 2020, Moon et al., 29 Aug 2025, Liu et al., 2012, Muta et al., 28 Nov 2025). A plausible implication is that the construction remains attractive precisely because it is structurally rigid enough to admit exact formulas, yet flexible enough to support a large spectrum of variants.