Subdivision Vertex Neighbourhood Corona

• Subdivision vertex neighbourhood corona is a graph operation that constructs a composite graph by using the subdivision of one graph and attaching copies of a second graph through neighbouring vertices.
• It enables detailed spectral analysis by relating the spectra of the composite graph to those of the original graphs via constrained coronals and low-degree algebraic equations.
• Applications include cospectral graph construction, spanning tree enumeration, expander graph design, and extensions to signed graphs and chromatic invariants.

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$0 (Liu et al., 2012). 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 $G_2$1-chromatic number (Liu et al., 2012, Lu et al., 2013, Rajkumar et al., 2020, Guragain et al., 2023, Falcón et al., 2023).

1. Definition and basic construction

Let $G_2$2 and $G_2$3 be vertex-disjoint graphs. The subdivision graph $G_2$4 is the graph obtained by inserting a new vertex into every edge of $G_2$5 (Liu et al., 2012, Lu et al., 2013). If $G_2$6 denotes the set of inserted vertices of $G_2$7, then the subdivision-vertex neighbourhood corona $G_2$8 is formed from $G_2$9 and $G_1 \boxdot G_2$0 copies of $G_1 \boxdot G_2$1, all vertex disjoint, by joining the neighbours of the $G_1 \boxdot G_2$2-th vertex of $G_1 \boxdot G_2$3 to every vertex in the $G_1 \boxdot G_2$4-th copy of $G_1 \boxdot G_2$5 (Liu et al., 2012).

If $G_1 \boxdot G_2$6 has $G_1 \boxdot G_2$7 vertices and $G_1 \boxdot G_2$8 edges, and $G_1 \boxdot G_2$9 has $\mathcal{S}(G_1)$0 vertices and $\mathcal{S}(G_1)$1 edges, then $\mathcal{S}(G_1)$2 has $\mathcal{S}(G_1)$3 vertices and $\mathcal{S}(G_1)$4 edges (Liu et al., 2012). In the notation used for $\mathcal{S}(G_1)$5, the vertex set consists of the original vertices $\mathcal{S}(G_1)$6, the subdivision vertices $\mathcal{S}(G_1)$7, and for each $\mathcal{S}(G_1)$8, a copy $\mathcal{S}(G_1)$9 with vertices $|V(G_1)|$0 (Falcón et al., 2023).

A degree formula is available in the $|V(G_1)|$1 notation. For $|V(G_1)|$2,

$|V(G_1)|$3

(Falcón et al., 2023). This makes explicit that the original vertices retain their degrees from $|V(G_1)|$4, 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 $|V(G_1)|$5 is obtained from $|V(G_1)|$6 and $|V(G_1)|$7 copies of $|V(G_1)|$8 by joining the $|V(G_1)|$9-th vertex of $G_2$0 itself to every vertex in the $G_2$1-th copy of $G_2$2 (Lu et al., 2013). Thus $G_2$3 attaches copies directly to original vertices, whereas $G_2$4 attaches through their neighbours in the subdivision graph (Liu et al., 2012, Lu et al., 2013).

The distinction from the subdivision vertex join is also explicit. In the subdivision vertex join $G_2$5, each subdivision vertex added to $G_2$6 in $G_2$7 is joined to every vertex of $G_2$8 (Tabish et al., 2021). This is a global connection pattern. By contrast, the subdivision-vertex neighbourhood corona uses $G_2$9 copies of the second graph and localizes the attachment to the neighbours of each original vertex in the subdivision graph (Liu et al., 2012). The available summary further states that neither corona join nor subdivision vertex join is exactly the subdivision vertex neighbourhood corona (Tabish et al., 2021).

A unifying interpretation is provided by the generalized corona constrained by vertex subsets. Given a base graph $i$0, graphs $i$1, and subsets $i$2 with $i$3, the generalized corona $i$4 is formed by joining $i$5 to all vertices in $i$6 (Rajkumar et al., 2020). The subdivision vertex neighbourhood corona is obtained by taking $i$7 and $i$8 for all $i$9 (Rajkumar et al., 2020). 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 $V(G_1)$0. For regular $V(G_1)$1, these spectra are expressed in terms of the spectra of $V(G_1)$2 and $V(G_1)$3, together with the coronal $V(G_1)$4, defined as the sum of all entries of $V(G_1)$5 (Liu et al., 2012).

For the adjacency matrix, if $V(G_1)$6 is $V(G_1)$7-regular of order $V(G_1)$8 and size $V(G_1)$9, and $\mathcal{S}(G_1)$0 is arbitrary of order $\mathcal{S}(G_1)$1, then

$\mathcal{S}(G_1)$2

(Liu et al., 2012). When $\mathcal{S}(G_1)$3 is $\mathcal{S}(G_1)$4-regular, the coronal simplifies to

$\mathcal{S}(G_1)$5

and a cubic equation yields part of the spectrum (Liu et al., 2012).

For the Laplacian matrix,

$\mathcal{S}(G_1)$6

(Liu et al., 2012).

For the signless Laplacian matrix,

$\mathcal{S}(G_1)$7

(Liu et al., 2012).

These formulas show that the operation is spectrally tractable despite its composite structure. In particular, inherited factors from $\mathcal{S}(G_1)$8 appear with multiplicity $\mathcal{S}(G_1)$9, while new eigenvalues arise through low-degree equations depending on the eigenvalues of $i$0 (Liu et al., 2012).

The generalized-corona framework provides a second spectral derivation. For $i$1, the adjacency characteristic polynomial is

$i$2

with

$i$3

where $i$4 is the coronal of a matrix constrained by an index set $i$5 (Rajkumar et al., 2020). For the subdivision vertex neighbourhood corona, $i$6 and $i$7 (Rajkumar et al., 2020). If $i$8 is $i$9-regular with $G_2$00, then

$G_2$01

(Rajkumar et al., 2020). The corresponding adjacency polynomial becomes

$G_2$02

(Rajkumar et al., 2020).

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_2$03 and $G_2$04 are, for example, adjacency-cospectral regular graphs, then for any $G_2$05, the graphs $G_2$06 and $G_2$07 are cospectral; if $G_2$08 and $G_2$09 are cospectral with equal coronal and $G_2$10 is regular, then $G_2$11 and $G_2$12 are cospectral (Liu et al., 2012). The same pattern extends to Laplacian and signless Laplacian spectra (Liu et al., 2012).

The Laplacian spectrum also yields a closed form for the number of spanning trees. For $G_2$13,

$G_2$14

(Liu et al., 2012).

A further application concerns expander graph construction. If $G_2$15 is a family of $G_2$16-regular expanders and $G_2$17 is an edgeless graph with $G_2$18 vertices, then iterated application of the $G_2$19 operation produces a new expander family, with algebraic connectivity tracked by

$G_2$20

(Liu et al., 2012).

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 (Rajkumar et al., 2020). In this framework, the key innovation is the coronal constrained by an index set,

$G_2$21

where $G_2$22 is the indicator vector for $G_2$23 (Rajkumar et al., 2020). 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 (Rajkumar et al., 2020). 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 $G_2$24 and $G_2$25, the subdivision vertex neighbourhood corona is defined using an $G_2$26-orientation $G_2$27 of $G_2$28 (Guragain et al., 2023). The construction inserts a new vertex into every edge, takes $G_2$29 copies of $G_2$30, and for the $G_2$31-th vertex $G_2$32 of $G_2$33, connects each of its neighbors within the subdivision graph to every vertex in the $G_2$34-th copy of $G_2$35; the sign of an edge between a neighbor $G_2$36 of $G_2$37 and a vertex $G_2$38 in the $G_2$39-th copy is $G_2$40 (Guragain et al., 2023).

The associated spectral theory uses the signed coronal

$G_2$41

(Guragain et al., 2023). For a $G_2$42-regular signed graph $G_2$43, the adjacency characteristic polynomial is

$G_2$44

and analogous formulas are given for the Laplacian, signless Laplacian, and normalized Laplacian matrices (Guragain et al., 2023). The paper further states that these formulas remain valid for all $G_2$45-orientations and that the resulting spectral properties are robust under switching equivalence (Guragain et al., 2023).

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 $G_2$46-chromatic number. For a graph $G_2$47, the $G_2$48-chromatic number $G_2$49 is the largest integer $G_2$50 such that there is a proper $G_2$51-coloring with a $G_2$52-vertex for each color (Falcón et al., 2023). The general bounds

$G_2$53

are used in the analysis of $G_2$54 (Falcón et al., 2023).

For the subdivision-vertex neighbourhood corona, the maximum degree satisfies

$G_2$55

(Falcón et al., 2023). If $G_2$56, then:

1. $G_2$57,
2. if $G_2$58, then $G_2$59 (Falcón et al., 2023).

The paper determines $G_2$60 when $G_2$61 and $G_2$62 are paths, cycles, or stars, and also for several cases involving complete graphs (Falcón et al., 2023). Selected exact formulas include:

• For $G_2$63,

$G_2$64

(Falcón et al., 2023).

• For cycles and paths,

$G_2$65

(Falcón et al., 2023).

• For stars with complete graphs,

$G_2$66

(Falcón et al., 2023).

The same source records that the $G_2$67-chromatic number of $G_2$68, the subdivision graph of $G_2$69, is $G_2$70 if $G_2$71 is odd, contradicting some prior claims in the literature (Falcón et al., 2023). 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_2$72, obtained by indexing copies of $G_2$73 by subdivision vertices rather than original vertices (Liu et al., 2012). 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 (Liu et al., 2018).

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 (Cavers et al., 20 May 2026). 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$74, producing a more global connection pattern (Tabish et al., 2021). In the subdivision-vertex neighbourhood corona