---
title: Subdivision Vertex Neighbourhood Corona
url: https://www.emergentmind.com/topics/subdivision-vertex-neighbourhood-corona
type: topic
---

# Subdivision Vertex Neighbourhood Corona

Searching arXiv for recent and foundational papers on subdivision vertex neighbourhood corona.
The **subdivision-vertex neighbourhood corona** is a graph operation defined for two vertex-disjoint graphs \(G_1\) and \(G_2\). Denoted \(G_1 \boxdot G_2\), it is obtained from the subdivision graph \(\mathcal{S}(G_1)\) and \(|V(G_1)|\) copies of \(G_2\), by joining the neighbours of the \(i\)-th vertex of \(V(G_1)\) in \(\mathcal{S}(G_1)\) to every vertex in the \(i\)-th copy of \(G_2\) [1212.0851]. This operation belongs to the broader family of corona-type graph products, but it is structurally distinct from the standard corona, the subdivision-vertex corona, and the subdivision vertex join. Its study has developed along several lines, including spectral graph theory, generalized corona frameworks, signed graph extensions, and coloring invariants such as the \(b\)-chromatic number [1212.0851] [1302.0457] [2008.05428] [2312.12845] [2302.13667].

## 1. Definition and basic construction

Let \(G_1\) and \(G_2\) be vertex-disjoint graphs. The subdivision graph \(\mathcal{S}(G_1)\) is the graph obtained by inserting a new vertex into every edge of \(G_1\) [1212.0851] [1302.0457]. If \(I(G_1)\) denotes the set of inserted vertices of \(\mathcal{S}(G_1)\), then the subdivision-vertex neighbourhood corona \(G_1 \boxdot G_2\) is formed from \(\mathcal{S}(G_1)\) and \(|V(G_1)|\) copies of \(G_2\), all vertex disjoint, by joining the neighbours of the \(i\)-th vertex of \(V(G_1)\) to every vertex in the \(i\)-th copy of \(G_2\) [1212.0851].

If \(G_1\) has \(n_1\) vertices and \(m_1\) edges, and \(G_2\) has \(n_2\) vertices and \(m_2\) edges, then \(G_1\boxdot G_2\) has \(n_1 + m_1 + n_1n_2\) vertices and \(2m_1 + n_1 m_2 + 2m_1 n_2\) edges [1212.0851]. In the notation used for \(G \boxdot H\), the vertex set consists of the original vertices \(V(G)\), the subdivision vertices \(I(G)\), and for each \(u_i \in V(G)\), a copy \(H_i\) with vertices \(v_{i,0},\ldots,v_{i,t-1}\) [2302.13667].

A degree formula is available in the \(G \boxdot H\) notation. For \(v \in V(G \boxdot H)\),
\[
d_{G \boxdot H}(v) = \begin{cases}
d_G(v), & v \in V(G)\\
2 \cdot |V(H)| + 2, & v \in I(G)\\
d_G(u_i) + d_H(v_j), & v = v_{i,j}
\end{cases}
\]
[2302.13667]. This makes explicit that the original vertices retain their degrees from \(G\), whereas the inserted subdivision vertices absorb the attachment load.

## 2. Position among related corona-type products

The subdivision-vertex neighbourhood corona is closely related to, but different from, several other subdivision-based products. The **subdivision-vertex corona** \(G_1 \odot G_2\) is obtained from \(\mathcal{S}(G_1)\) and \(|V(G_1)|\) copies of \(G_2\) by joining the \(i\)-th vertex of \(V(G_1)\) itself to every vertex in the \(i\)-th copy of \(G_2\) [1302.0457]. Thus \(G_1 \odot G_2\) attaches copies directly to original vertices, whereas \(G_1 \boxdot G_2\) attaches through their neighbours in the subdivision graph [1212.0851] [1302.0457].

The distinction from the **subdivision vertex join** is also explicit. In the subdivision vertex join \(G_1 \dotplus G_2\), each subdivision vertex added to \(G_1\) in \(S(G_1)\) is joined to every vertex of \(G_2\) [2111.11006]. This is a global connection pattern. By contrast, the subdivision-vertex neighbourhood corona uses \(|V(G_1)|\) copies of the second graph and localizes the attachment to the neighbours of each original vertex in the subdivision graph [1212.0851]. The available summary further states that neither corona join nor subdivision vertex join is exactly the subdivision vertex neighbourhood corona [2111.11006].

A unifying interpretation is provided by the generalized corona constrained by vertex subsets. Given a base graph \(G\), graphs \(\mathcal{H}=(H_1,\ldots,H_n)\), and subsets \(\mathcal{T}=(T_1,\ldots,T_n)\) with \(T_i \subseteq V(H_i)\), the generalized corona \(G \circledast_{\mathcal T}\mathcal H\) is formed by joining \(v_i \in G\) to all vertices in \(T_i\subseteq V(H_i)\) [2008.05428]. The subdivision vertex neighbourhood corona is obtained by taking \(H_i=S(H)\) and \(T_i=V(H)\) for all \(i\) [2008.05428]. In this sense, the operation is a special case of a larger constrained-corona formalism.

This suggests a structural characterization: the subdivision-vertex neighbourhood corona interpolates between purely local attachment rules and more global subdivision-based joins, and can be represented either as a standalone graph product or as an instance of a generalized subset-constrained corona.

## 3. Spectral theory

A major part of the literature concerns the adjacency, Laplacian, and signless Laplacian spectra of \(G_1 \boxdot G_2\). For regular \(G_1\), these spectra are expressed in terms of the spectra of \(G_1\) and \(G_2\), together with the coronal \(\Gamma_M(x)\), defined as the sum of all entries of \((xI-M)^{-1}\) [1212.0851].

For the adjacency matrix, if \(G_1\) is \(r_1\)-regular of order \(n_1\) and size \(m_1\), and \(G_2\) is arbitrary of order \(n_2\), then
\[
\phi(A(G_1\boxdot G_2);x) = x^{m_1-n_1} \cdot \big(\phi(A(G_2);x)\big)^{n_1} \cdot \prod_{i=1}^{n_1}\left(x^2 - [1 + x \Gamma_{A(G_2)}(x)](\lambda_i(G_1)+r_1)\right)
\]
[1212.0851]. When \(G_2\) is \(r_2\)-regular, the coronal simplifies to
\[
\Gamma_{A(G_2)}(x) = \frac{n_2}{x-r_2},
\]
and a cubic equation yields part of the spectrum [1212.0851].

For the Laplacian matrix,
\[
\begin{aligned}
\phi(L(G_1\boxdot G_2); x) =\quad & x(x-2-2n_2-r_1)(x-2-2n_2)^{m_1-n_1}\cdot\\
& \prod_{i=1}^{n_2}(x - r_1 - \mu_i(G_2))^{n_1}\cdot\\
& \prod_{i=2}^{n_1}\left(x^2 - (2 + 2n_2 + r_1)x + (1 + n_2)\mu_i(G_1)\right)
\end{aligned}
\]
[1212.0851].

For the signless Laplacian matrix,
\[
\begin{aligned}
\phi(Q(G_1\boxdot G_2); x) =\quad & (x-2-2n_2)^{m_1-n_1} \cdot \prod_{i=1}^{n_2}(x - r_1 - \nu_i(G_2))^{n_1}\\
& \cdot\prod_{i=1}^{n_1} \big((x-2-2n_2)(x - r_1) - [1 + (x - r_1)\Gamma_{Q(G_2)}(x - r_1)]\nu_i(G_1) \big)
\end{aligned}
\]
[1212.0851].

These formulas show that the operation is spectrally tractable despite its composite structure. In particular, inherited factors from \(G_2\) appear with multiplicity \(n_1\), while new eigenvalues arise through low-degree equations depending on the eigenvalues of \(G_1\) [1212.0851].

The generalized-corona framework provides a second spectral derivation. For \(G\circledast_{\mathcal T}\mathcal H\), the 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
\[
U_A = \textrm{diag}\left( \Gamma_{H_1}^{T_1}(x), \ldots, \Gamma_{H_n}^{T_n}(x) \right),
\]
where \(\Gamma_{M}^{\alpha}(x)\) is the coronal of a matrix constrained by an index set \(\alpha\) [2008.05428]. For the subdivision vertex neighbourhood corona, \(H_i=S(H)\) and \(T_i=V(H)\) [2008.05428]. If \(H\) is \(r\)-regular with \(h=|V(H)|\), then
\[
\Gamma_{S(H)}^{V(H)}(x) = \frac{h x}{x^2 - 2r}
\]
[2008.05428]. The corresponding adjacency polynomial becomes
\[
P_{\textrm{SVNC}}(x) = \big(P_{S(H)}(x)\big)^n \cdot \left| xI_n - A(G) - \Gamma_{S(H)}^{V(H)}(x) \cdot I_n \right|
\]
[2008.05428].

A plausible implication is that the constrained-coronal formalism isolates the exact spectral effect of joining to a selected vertex subset—here, the original vertices of a subdivision graph—rather than to all vertices.

## 4. Applications in spectral graph theory

The spectral formulas support several downstream constructions. One such application is the construction of cospectral graphs. If \(G_1\) and \(G_2\) are, for example, adjacency-cospectral regular graphs, then for any \(H\), the graphs \(G_1 \boxdot H\) and \(G_2 \boxdot H\) are cospectral; if \(H_1\) and \(H_2\) are cospectral with equal coronal and \(G\) is regular, then \(G \boxdot H_1\) and \(G \boxdot H_2\) are cospectral [1212.0851]. The same pattern extends to Laplacian and signless Laplacian spectra [1212.0851].

The Laplacian spectrum also yields a closed form for the number of spanning trees. For \(G_1 \boxdot G_2\),
\[
t(G_1 \boxdot G_2) = \frac{(2 + 2n_2 + r_1)(2 + 2n_2)^{m_1 - n_1} \prod_{i=1}^{n_2}(r_1 + \mu_i(G_2))^{n_1} \prod_{i=2}^{n_1}(1 + n_2)\mu_i(G_1)}{n_1 + m_1 + n_1 n_2}
\]
[1212.0851].

A further application concerns expander graph construction. If \(\mathcal{G}=\{G_i\}\) is a family of \(r\)-regular expanders and \(H\) is an edgeless graph with \(\frac{r}{2}-1\) vertices, then iterated application of the \( \boxdot \) operation produces a new expander family, with algebraic connectivity tracked by
\[
f(x) = r - \sqrt{r^2 - \frac{r}{2}x}
\]
[1212.0851].

These results place the subdivision-vertex neighbourhood corona among graph products that are not only spectrally analyzable but also operationally useful for generating examples with controlled spectral behavior.

## 5. Generalizations and variants

The generalized corona of graphs constrained by vertex subsets subsumes the subdivision vertex neighbourhood corona as a special case [2008.05428]. In this framework, the key innovation is the coronal constrained by an index set,
\[
\Gamma_{M}^{\alpha}(x) = \mathbf{r}_\alpha (xI_n - M)^{-1} \mathbf{r}_\alpha^T,
\]
where \(\mathbf{r}_\alpha\) is the indicator vector for \(\alpha\) [2008.05428]. The adjacency, Laplacian, and signless Laplacian characteristic polynomials of the generalized corona are then expressed in terms of constrained coronals and characteristic polynomials of the constituent graphs [2008.05428]. The subdivision vertex neighbourhood corona is one of the principal examples motivating this formulation.

The signed-graph extension introduces additional algebraic structure. For signed graphs \(\Gamma^1=(G_1,\sigma_1,\mu_1)\) and \(\Gamma^2=(G_2,\sigma_2,\mu_2)\), the subdivision vertex neighbourhood corona is defined using an \(r\)-orientation \(\theta\) of \(\Gamma^1\) [2312.12845]. The construction inserts a new vertex into every edge, takes \(n_1=|V(G_1)|\) copies of \(\Gamma^2\), and for the \(t\)-th vertex \(v_t\) of \(G_1\), connects each of its neighbors within the subdivision graph to every vertex in the \(t\)-th copy of \(\Gamma^2\); the sign of an edge between a neighbor \(v_e\) of \(v_t\) and a vertex \(u\) in the \(t\)-th copy is \(\theta(v_t,e)\mu_2(u)\) [2312.12845].

The associated spectral theory uses the signed coronal
\[
\Sigma_N(X) = \mu(\Gamma)^T (X I_n - N)^{-1} \mu(\Gamma)
\]
[2312.12845]. For a \(\gamma_1\)-regular signed graph \(\Gamma^1\), the adjacency characteristic polynomial is
\[
f_{A((\Gamma^1 \boxdot \Gamma^2)_\theta)}(\lambda) = \lambda^{m_1-n_1} \left[ f_{A(\Gamma^2)}(\lambda) \right]^{n_1}  \prod_{j=1}^{n_1} \left[ \lambda^2 - (1+\lambda \Sigma_{A(\Gamma^2)}(\lambda)) (\lambda_j + \gamma_1) \right]
\]
and analogous formulas are given for the Laplacian, signless Laplacian, and normalized Laplacian matrices [2312.12845]. The paper further states that these formulas remain valid for all \(r\)-orientations and that the resulting spectral properties are robust under switching equivalence [2312.12845].

A plausible interpretation is that the signed generalization preserves the combinatorial skeleton of the unsigned operation while replacing ordinary coronals with signed coronals and adjacency-based attachments with sign-consistent ones.

## 6. Coloring and combinatorial invariants

Beyond spectra, the subdivision-vertex neighbourhood corona has been studied through the \(b\)-chromatic number. For a graph \(G\), the \(b\)-chromatic number \(\varphi(G)\) is the largest integer \(k\) such that there is a proper \(k\)-coloring with a \(b\)-vertex for each color [2302.13667]. The general bounds
\[
\chi(G) \leq \varphi(G) \leq m(G) \leq \Delta(G) + 1
\]
are used in the analysis of \(G \boxdot H\) [2302.13667].

For the subdivision-vertex neighbourhood corona, the maximum degree satisfies
\[
\Delta(G \boxdot H) = \max\{2|V(H)| + 2,\, \Delta(G) + \Delta(H)\}
\]
[2302.13667]. If \(\Delta(G) \leq |V(H)| + 3\), then:
1. \(\varphi(G \boxdot H) \leq 2|V(H)| + 3\),
2. if \(\Delta(G) + \Delta(H) + 1 \leq |I(G)| \leq 2|V(H)| + 2\), then \(\varphi(G \boxdot H) \leq |I(G)|\)
[2302.13667].

The paper determines \(\varphi(G\boxdot H)\) when \(G\) and \(H\) are paths, cycles, or stars, and also for several cases involving complete graphs [2302.13667]. Selected exact formulas include:

- For \(t>2\),
\[
\varphi(P_n\boxdot P_t) = \begin{cases}
4, & n = 3, t \in \{3,4\} \\
5, & n = 3, t \geq 5 \text{ or } n \in \{4,5\} \\
n-1, & 6 \leq n \leq 2t + 3 \\
2t + 3, & n > 2t + 3
\end{cases}
\]
[2302.13667].

- For cycles and paths,
\[
\varphi(C_n \boxdot P_t) = \varphi(C_n \boxdot C_t) =
\begin{cases}
5, & n \in \{3,4\} \\
n, & 5 \leq n \leq 2t+2 \\
2t+3, & n > 2t+2
\end{cases}
\]
[2302.13667].

- For stars with complete graphs,
\[
\varphi(S_n\boxdot K_{t'}) = \min\{n, t'+2\} + t'
\]
[2302.13667].

The same source records that the \(b\)-chromatic number of \(S(K_n)\), the subdivision graph of \(K_n\), is \(n\) if \(n\) is odd, contradicting some prior claims in the literature [2302.13667]. This places the subdivision-vertex neighbourhood corona within a combinatorial program extending beyond eigenvalue computations.

## 7. Broader context and related directions

The subdivision-vertex neighbourhood corona is part of an active family of neighbourhood-based and subdivision-based graph products. The original spectral study paired it with the subdivision-edge neighbourhood corona \(G_1 \boxminus G_2\), obtained by indexing copies of \(G_2\) by subdivision vertices rather than original vertices [1212.0851]. Later work on generalized subdivision-coronae examined normalized Laplacian spectra for related operations such as generalized subdivision-vertex corona and generalized subdivision-edge corona, emphasizing how the choice of attachment set changes multiplicities and the algebraic equations determining additional eigenvalues [1807.03939].

More recent work on digraphs extends the subdivision-vertex corona and related subdivision products, but explicitly notes that the subdivision-vertex neighbourhood corona is mentioned as previously defined for undirected graphs and is not directly defined for digraphs in that paper [2605.21775]. This indicates that the neighbourhood version remains primarily an undirected, and more recently signed, construction.

The relation to the subdivision vertex join is particularly important for avoiding terminological confusion. In the join construction, every subdivision vertex is joined to all vertices of \(G_2\), producing a more global connection pattern [2111.11006]. In the subdivision-vertex neighbourhood corona

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