---
title: Vertex-Corona in Graph Theory
url: https://www.emergentmind.com/topics/vertex-corona
type: topic
---

# Vertex-Corona in Graph Theory

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 [1503.07842], [1204.0647], [2509.14481], [2403.02650].

## 1. Definition and notation

Let \(G_1\) and \(G_2\) be graphs on disjoint vertex sets, with \(|V(G_1)|=n_1\) and \(|V(G_2)|=n_2\). The corona of \(G_1\) and \(G_2\), denoted in the cited literature by either \(G_1\circ G_2\) or \(G_1\odot G_2\), is obtained from one copy of \(G_1\) and \(n_1\) copies of \(G_2\) by joining the \(i\)-th vertex of \(G_1\) to every vertex in the \(i\)-th copy of \(G_2\) [1503.07842], [1204.0647]. In set-theoretic form,
\[
V(G\circ H)=V(G)\cup \bigcup_{g\in V(G)}V(H_g),
\]
where \(H_g\) denotes the copy attached to \(g\) [2002.07451].

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 \(G\circ H\) and \(G\odot H\), while other symbols are overloaded for different operations. For example, one paper uses \(\diamond\) for neighborhood corona, whereas another uses \(\diamond\) for edge corona [1503.07842], [2002.07451]. 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 [1611.04219].

The elementary combinatorial effect of the operation is immediate in special families. For the cyclic base \(C_n\) and complete attachment \(K_p\), the graph \(C_n\odot K_p\) has \(n(p+1)\) vertices, and the analogous double corona \(C_n K_p\) has \(n(p+2)\) vertices [2508.21764]. This block structure underlies most later exact formulas.

## 2. Linear-algebraic formulation

The Laplacian description of the vertex-corona is particularly tractable when \(G_1\) is \(r_1\)-regular and \(G_2\) is arbitrary. With a suitable labeling, the Laplacian of \(G_1\circ G_2\) has the block form
\[
L(G_1 \circ G_2) =
\begin{pmatrix}
L(G_1) + n_2 I_{n_1} & -\mathbf{1}_{n_2} \otimes I_{n_1} \\
-(\mathbf{1}_{n_2} \otimes I_{n_1})^\top & (L(G_2) + I_{n_2}) \otimes I_{n_1}
\end{pmatrix},
\]
where \(\mathbf{1}_{n_2}\) is the all-1 vector and \(\otimes\) is the Kronecker product [1503.07842]. This decomposition makes the attachment mechanism explicit: the base-graph block is shifted by \(n_2 I_{n_1}\), 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:
\[
\chi(G\odot H)=\max\{\chi(G),\chi(H)+1\}.
\]
The additional \(+1\) reflects the fact that the central vertex of each corona block must avoid the colors used in its attached copy [1204.0647].

The operation also admits flexible generalizations in which the attachment is constrained to prescribed subsets. If \(G\) has vertices \(v_1,\dots,v_n\), \(\mathcal H=(H_1,\dots,H_n)\), and each \(T_i\subseteq V(H_i)\), then the generalized corona constrained by vertex subsets joins \(v_i\) only to the vertices of \(T_i\). Its adjacency characteristic polynomial is
\[
P_{G \circledast_{\mathcal T}\mathcal H}(x)
=
\left[\prod_{i=1}^{n} P_{H_i}(x)\right]\cdot
\left|xI_n-A(G)-U_A\right|,
\]
with
\[
U_A=\operatorname{diag}\big(\Gamma_{H_1}^{T_1}(x),\dots,\Gamma_{H_n}^{T_n}(x)\big),
\]
where \(\Gamma_M^\alpha(x)=\mathbf r_\alpha (xI-M)^{-1}\mathbf r_\alpha^T\) is the coronal constrained by an index set [2008.05428]. 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 \(\{1\}\)-inverse \(L^+\) of a graph Laplacian,
\[
r_{ij}=(L^+)_{ii}+(L^+)_{jj}-2(L^+)_{ij},
\]
which is Klein and Randić’s formula for resistance distance [1503.07842]. For \(G_1\circ G_2\), a symmetric \(\{1\}\)-inverse is obtained from the block quantities
\[
L_1=L(G_1)+n_2I_{n_1},\qquad
L_2=-\mathbf1_{n_2}\otimes I_{n_1},\qquad
L_3=(L(G_2)+I_{n_2})\otimes I_{n_1},
\]
and
\[
S=L_3-J_{n_2\times n_2}\otimes L_1^{-1},
\]
yielding
\[
\begin{pmatrix}
L_1^{-1}+L_1^{-1}L_2S^\#L_2^\top L_1^{-1} & -L_1^{-1}L_2S^\#\\
-S^\#L_2^\top L_1^{-1} & S^\#
\end{pmatrix}
\]
as a symmetric \(\{1\}\)-inverse of \(L(G_1\circ G_2)\) [1503.07842]. The explicit examples \(C_3\circ P_3\) and \(C_4\odot P_2\) 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 \(G_1\odot G_2\), the number of vertices is
\[
n_1(1+n_2+m_2),
\]
the number of edges is
\[
m_1+n_1n_2+2n_1m_2,
\]
and the resistance formulas are expressed through \(L_{G_1}^{\#}\), \(Q=L_{G_2}+2I_{n_2}\), and Kronecker products [1611.04219]. The same paper also gives a Kirchhoff-index formula in terms of \(Q^{-1}\), \(A_{G_2}\), \(D_{G_2}\), the Laplacian eigenvalues of \(G_2\), and \(Kf(G_1)\) [1611.04219]. 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 [1212.0851].

## 4. Coloring, domination, and game parameters

The vertex-corona is a prolific testbed for refined coloring parameters. For distance-\(k\) coloring, the cited literature gives, among others,
\[
\Delta_1+n_2+1 \le \chi_{\le 2}(G\odot H)\le \chi_{\le 2}(G)+n_2,
\]
\[
\chi_{\le 2}(P_{n_1}\odot H)=n_2+3 \quad (n_1\ge 3),
\]
and, for \(2\le k\le n_1\),
\[
\chi_{\le k}(P_{n_1}\odot H)=
\begin{cases}
n_2(k-1)+k+1,& k\le n_1-1,\\
n_2(k-1)+k,& k=n_1.
\end{cases}
\]
There are also exact formulas for trees and cycles in distance-2 and distance-3 settings [1204.0647].

For domination-related colorings, the corona product exhibits a sharp separation between dominated and dominator chromatic behavior. The dominated chromatic number satisfies
\[
\chi_{dom}(G\circ H)=n(G)\chi(H),
\]
whereas the dominator chromatic number satisfies
\[
\chi_d(G\circ H)=n(G)+\chi(H).
\]
For example,
\[
\chi_d(C_4\circ K_2)=4+2=6.
\]
These formulas show that the corona structure allows each central vertex of \(G\) to act as a singleton dominator class while reusing an optimal palette on the attached copies [2002.07451].

Classical domination parameters also admit exact corona formulas. If \(G\) has order \(n_1\) and \(H\) has order \(n_2\ge 2\), then
\[
\gamma_R(G\odot H)=2n_1.
\]
For \(k\ge 2\),
\[
\gamma_k(G\odot H)=
n\cdot \min\{\gamma_k(H),\gamma_{k-1}(H)+1\},
\]
and the distance-\(k\) domination number satisfies
\[
\gamma_{\le k}(G\odot H)=\gamma_{\le k-1}(G).
\]
Further exact formulas are given for independence domination, the domatic number, and the idomatic number [1204.0647].

The Maker-Breaker domination game on corona products is governed primarily by the second factor. For \(n(G)\ge 2\), if Dominator can win on \(H\) or if the first player can win on \(H\), then
\[
o(G\circ H)=D;
\]
if Staller can win on \(H\), then
\[
o(G\circ H)=S.
\]
When \(o(H)=S\), the sharp exact formula
\[
\gamma_{SMB}(G\circ H)=\gamma_{SMB}'(G\circ H)=1+\gamma_{SMB}(H)
\]
holds [2402.13581]. 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 \(k\)-threshold process on \(C_n\odot K_p\), the irreversible \(k\)-threshold conversion number is
\[
C_k(C_n \odot K_p)=
\begin{cases}
(k-1)n+1,& k\le p+1,\\
pn+\left\lceil \frac n2 \right\rceil,& k=p+2,\\
\text{\(k\)-inconvertible},& k\ge p+3.
\end{cases}
\]
The associated random-seeding success probabilities are likewise given explicitly, and the paper defines the resilience factor as
\[
\mathrm{RF}[\,C_k(C_n\odot K_p)\,]=1-P(S).
\]
A parallel theory is developed there for double corona product graphs \(C_nK_p\) [2508.21764]. 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 \(n\ge 2\) and \(m\ge 2\),
\[
\chi_{la}(P_n\circ \overline{K_m})=mn+2.
\]
For cycles, the behavior splits by parity: for even \(n\ge 4\) and any \(m\ge 2\),
\[
\chi_{la}(C_n\circ \overline{K_m})=mn+2,
\]
whereas for \(C_3\),
\[
\chi_{la}(C_3\circ \overline{K_m})=3m+3 \quad (m\ge 2).
\]
For complete graphs,
\[
\chi_{la}(K_n\circ \overline{K_m})=mn+n \quad (m\ge 2,\ n\ge 3),
\]
and, in the one-leaf case,
\[
\chi_{la}(K_n\circ K_1)=2n-1 \quad (n\ge 3).
\]
These formulas are derived by explicit edge labelings and lower-bound arguments based on leaf structure and vertex sums [1808.04956].

A correction to earlier lower-bound proofs was later supplied for friendship and fan graph coronas with null graphs. In particular, for \(n\ge 2\),
\[
\chi_{la}(f_n\circ O_1)=2n+3,
\]
and the lower bounds
\[
\chi_{la}(f_n\circ O_m)\ge m(2n+1)+3 \quad (m\ge 2),
\qquad
\chi_{la}(F_n\circ O_m)\ge m(n+1)+3 \quad (n\ge 3)
\]
were established with complete proofs [2203.09906].

## 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, \(Q\)-vertex corona, \(T\)-vertex neighborhood corona, and the subdivision vertex-edge neighbourhood vertex-corona [1503.07842], [1212.0851], [2406.07183], [2509.12342], [1806.10133]. 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 \(Q(G)\) or \(T(G)\).

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 [2312.12843]. For digraphs, the symmetric-vertex-corona \(D_1\overleftrightarrow{\circ} D_2\) has adjacency matrix
\[
A(D_1 \overleftrightarrow{\circ} D_2)=
\begin{bmatrix}
A(D_1) & \mathbf{1}_{n_2}^T\otimes I_{n_1}\\
\mathbf{1}_{n_2}\otimes I_{n_1} & A(D_2)\otimes I_{n_1}
\end{bmatrix},
\]
and its adjacency, Laplacian, and signless Laplacian characteristic polynomials are expressed through digraph coronals [2509.14481]. For hypergraphs, the corona \(G_0^*\odot_1^n G_1^*\) generalizes the graph vertex-corona, and its adjacency and Seidel spectra are determined when \(G_1^*\) is regular [2403.02650].

The locating-chromatic number has also been analyzed in the ordinary corona setting. If \(G\) is connected and \(H\) has components \(H_1,\dots,H_k\), then
\[
\max_{1\le t\le k}\chi_L(H_t+K_1)
\le
\chi_L(G\odot H)
\le
\chi_L(G)+\sum_{t=1}^k\big[\chi_L(H_t+K_1)-1\big],
\]
with sharpness examples for both bounds [2408.06746]. 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
\[
\operatorname{reg}(S/I(G\circ \mathcal H))=\operatorname{im}(G\circ \mathcal H)
=
|\{\text{nontrivial attached cliques}\}|
\]
and
\[
\operatorname{pd}(S/I(G\circ \mathcal H))=\operatorname{bight}(I(G\circ \mathcal H)).
\]
The vertex-corona appears there as the case where all attached graphs are \(K_1\) [2511.22953].

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 [1503.07842], [2002.07451], [2508.21764], [1212.0851], [2511.22953]. 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.

Source: https://www.emergentmind.com/topics/vertex-corona