Neighbourhood Corona in Spectral Graph Theory

• Neighbourhood Corona is a graph operation that forms one copy of a base graph and multiple copies of a secondary graph, connecting each vertex’s neighbours to its corresponding copy.
• The construction leads to explicit block matrix formulations and spectral factorizations using Schur complements that relate the spectra of the product with those of its factors.
• Extensions to signed graphs, subdivisions, and T-graphs demonstrate its applications in cospectral constructions, expander properties, automorphism groups, and graph coloring parameters.

The neighbourhood corona of two graphs, denoted $G_{1}\star G_{2}$, is the graph obtained by taking one copy of $G_{1}$ and $|V(G_{1})|$ copies of $G_{2}$, and joining the neighbours of the $i$th vertex of $G_{1}$ to every vertex in the $i$th copy of $G_{2}$. In the literature on spectral graph theory, this operation is studied through adjacency, Laplacian, and signless Laplacian matrices, and it has been extended to signed graphs, subdivision-based constructions, and $T$-graph analogues. These developments connect neighbourhood coronae to cospectrality, expander constructions, automorphism groups, distinguishing invariants, and coloring parameters (Liu et al., 2012, Alikhani et al., 2016, Shamsher et al., 2023, Liu et al., 2012).

1. Definition and local structure

Let $G_{1}=(V(G_{1}),E(G_{1}))$ be a graph on $G_{1}$0 vertices $G_{1}$1, and let $G_{1}$2 be a graph on $G_{1}$3 vertices. For each $G_{1}$4, one forms the $G_{1}$5th copy of $G_{1}$6, and then joins every neighbour of $G_{1}$7 in $G_{1}$8 to every vertex in that $G_{1}$9th copy. The resulting degree formulas are

$|V(G_{1})|$0

which already show the asymmetry between the original $|V(G_{1})|$1-vertices and the vertices lying in the copies of $|V(G_{1})|$2 (Liu et al., 2012).

Several related constructions use the same neighbourhood-attachment principle but change the ambient graph or the attachment rule. In the signed setting, a signed graph is a pair $|V(G_{1})|$3, where $|V(G_{1})|$4 maps edges to $|V(G_{1})|$5, and the signed $|V(G_{1})|$6-neighbourhood corona $|V(G_{1})|$7 is obtained by joining every neighbour of the $|V(G_{1})|$8th vertex of $|V(G_{1})|$9 to every vertex in the $G_{2}$0th copy of $G_{2}$1 with the same sign as the corresponding incident edge in $G_{2}$2. In subdivision-based variants, the operation is performed after replacing each edge of $G_{2}$3 by a two-edge path. In $G_{2}$4-graph variants, the base graph is $G_{2}$5, whose vertices are the vertices and edges of $G_{2}$6 (Shamsher et al., 2023, Liu et al., 2012, Mukherjee et al., 15 Sep 2025).

Variant	Notation	Construction cue
Neighbourhood corona	$G_{2}$7	One copy of $G_{2}$8, $G_{2}$9 copies of $i$0, neighbours of $i$1 joined to the $i$2th copy
Signed $i$3-neighbourhood corona	$i$4	Same pattern, with new edges carrying the sign of the relevant incident edge
Subdivision-vertex neighbourhood corona	$i$5	Built from $i$6 and $i$7 copies of $i$8
Subdivision-edge neighbourhood corona	$i$9	Built from $G_{1}$0 and $G_{1}$1 copies of $G_{1}$2
$G_{1}$3-vertex / $G_{1}$4-edge neighbourhood corona	$G_{1}$5, $G_{1}$6	Built from $G_{1}$7, attaching copies at vertex- or edge-vertices

2. Block matrices, coronals, and Schur complements

A defining feature of neighbourhood coronae is that their matrices admit explicit block forms. For the ordinary neighbourhood corona, if the vertices are ordered by first listing $G_{1}$8 and then the $G_{1}$9 copies of $i$0, the adjacency matrix is

$i$1

The key auxiliary object is the coronal

$i$2

which enters after applying the Schur-complement formula to the lower-right block. In the signed case, the same strategy yields an analogous block decomposition for $i$3, with the off-diagonal blocks determined by the signed adjacency of $i$4 (Liu et al., 2012, Shamsher et al., 2023).

Subdivision-based neighbourhood coronae replace the adjacency blocks involving $i$5 by incidence-matrix blocks. For example, in the subdivision-vertex neighbourhood corona $i$6, the matrix formulas are written in terms of the vertex-edge incidence matrix $i$7 of $i$8. The same pattern persists in normalized Laplacian and $i$9-graph variants: the large matrix is partitioned into natural blocks, a Schur complement eliminates the repeated copies, and the remaining determinant factors into terms depending only on the spectra or coronals of the factors. This suggests a unifying methodological theme across the literature: block determinant reduction is the principal mechanism by which the spectra of neighbourhood coronae are made explicit (Liu et al., 2012, Wen et al., 2018, Mukherjee et al., 15 Sep 2025).

3. Spectra of the ordinary neighbourhood corona

The basic adjacency theorem of Liu and Zhou expresses the characteristic polynomial of $G_{2}$0 directly in terms of the adjacency spectrum of $G_{2}$1 and the $G_{2}$2-coronal of $G_{2}$3: $G_{2}$4 Here $G_{2}$5 are the eigenvalues of $G_{2}$6. When $G_{2}$7 is $G_{2}$8-regular, parallel formulas hold for the signless Laplacian and Laplacian characteristic polynomials, with $G_{2}$9 and $T$0 replacing the adjacency coronal. Since each row-sum of $T$1 is $T$2, one has $T$3, which yields explicit Laplacian eigenvalue formulas (Liu et al., 2012).

These factorisations have two immediate uses. First, they produce cospectral constructions. If $T$4 and $T$5 are $T$6-cospectral and $T$7 is any graph, then $T$8 and $T$9 are $G_{1}=(V(G_{1}),E(G_{1}))$0-cospectral. Likewise, if $G_{1}=(V(G_{1}),E(G_{1}))$1 are $G_{1}=(V(G_{1}),E(G_{1}))$2-cospectral regular graphs, then $G_{1}=(V(G_{1}),E(G_{1}))$3 and $G_{1}=(V(G_{1}),E(G_{1}))$4 are $G_{1}=(V(G_{1}),E(G_{1}))$5-cospectral. Second, the Laplacian formulas support expander constructions. If $G_{1}=(V(G_{1}),E(G_{1}))$6 is a family of non-complete $G_{1}=(V(G_{1}),E(G_{1}))$7-regular $G_{1}=(V(G_{1}),E(G_{1}))$8-expanders, and $G_{1}=(V(G_{1}),E(G_{1}))$9 is a fixed connected $G_{1}$00-regular graph with $G_{1}$01 even, then after adding edges so as to make each $G_{1}$02 $G_{1}$03-regular, the resulting family $G_{1}$04 is a family of $G_{1}$05-expanders (Liu et al., 2012).

4. Signed neighbourhood coronae

In signed graph theory, two different neighbourhood-corona constructions have been studied. The first is the signed $G_{1}$06-neighbourhood corona $G_{1}$07, where $G_{1}$08 is a signed graph and the new edges inherit the sign of the corresponding incident edge of $G_{1}$09. If $G_{1}$10 is $G_{1}$11-net-regular with adjacency eigenvalues $G_{1}$12, with $G_{1}$13, and $G_{1}$14, then the adjacency spectrum of $G_{1}$15 consists of: each $G_{1}$16, $G_{1}$17, repeated $G_{1}$18 times, and for each $G_{1}$19 two new eigenvalues

$G_{1}$20

The same paper determines the Laplacian spectrum for regular $G_{1}$21 and regular and net-regular $G_{1}$22, and the net Laplacian spectrum for net-regular $G_{1}$23 and arbitrary $G_{1}$24. It also identifies a “copy-eigenvalue” phenomenon: each eigenvalue of the second factor, except a distinguished pole, appears with multiplicity $G_{1}$25, while the remaining $G_{1}$26 eigenvalues arise from quadratic or rational equations. As consequences, the authors obtain signed graphs with $G_{1}$27 and $G_{1}$28 distinct adjacency, Laplacian and net Laplacian eigenvalues, and show that the signed neighbourhood corona of two signed graphs is not determined by its adjacency, Laplacian, or net Laplacian spectrum (Shamsher et al., 2023).

A second signed model, denoted $G_{1}$29, uses canonical vertex markings. If $G_{1}$30 and $G_{1}$31, then each new edge joining a neighbour $G_{1}$32 of $G_{1}$33 in $G_{1}$34 to a vertex $G_{1}$35 in the $G_{1}$36th copy of $G_{1}$37 is given sign

$G_{1}$38

Its adjacency characteristic polynomial is

$G_{1}$39

where $G_{1}$40. This construction also yields Laplacian and signless Laplacian formulas when $G_{1}$41 is regular. A notable structural result is that balanced factors do not by themselves guarantee a balanced product: even when both $G_{1}$42 and $G_{1}$43 are balanced, $G_{1}$44 fails to be balanced if and only if one of the factors contains one of three specified “bad” edge types (Sonar et al., 2023).

5. Automorphisms, distinguishing invariants, and coloring

The automorphism group of $G_{1}$45 is tightly controlled by the structure of the base graph. If $G_{1}$46 is connected and $G_{1}$47, then $G_{1}$48 restricts to an automorphism of $G_{1}$49, and the $G_{1}$50 copies of $G_{1}$51 are permuted among themselves. More precisely, every automorphism is uniquely determined by a pair

$G_{1}$52

where $G_{1}$53 acts on the central copy of $G_{1}$54 and each $G_{1}$55 acts within the $G_{1}$56th copy of $G_{1}$57 before that copy is moved according to $G_{1}$58 (Alikhani et al., 2016).

This automorphism description supports upper bounds on symmetry-breaking parameters. For connected $G_{1}$59 of order $G_{1}$60,

$G_{1}$61

More generally, if $G_{1}$62, $G_{1}$63, $G_{1}$64, $G_{1}$65, and for $G_{1}$66,

$G_{1}$67

then

$G_{1}$68

For the distinguishing index,

$G_{1}$69

for connected $G_{1}$70 with $G_{1}$71 (Alikhani et al., 2016).

Coloring theory enters through subdivision-based neighbourhood coronae. For graphs $G_{1}$72 and $G_{1}$73 that are paths, cycles, or stars, the subdivision-vertex neighbourhood corona $G_{1}$74 has explicitly determined $G_{1}$75-chromatic number, and the same is true for $G_{1}$76. The paper also establishes the $G_{1}$77-chromatic number for graphs $G_{1}$78 having $G_{1}$79-degree not greater than $G_{1}$80. Representative formulas include

$G_{1}$81

and

$G_{1}$82

The proofs combine upper bounds from $G_{1}$83 and $G_{1}$84 with explicit modular colorings and $G_{1}$85-rainbow sets (Falcón et al., 2023).

6. Subdivision, $G_{1}$86-graph, and normalized Laplacian generalizations

The subdivision graph $G_{1}$87 is obtained by inserting a new vertex into every edge of $G_{1}$88. On this basis, Liu and Lu defined the subdivision-vertex neighbourhood corona $G_{1}$89 and the subdivision-edge neighbourhood corona $G_{1}$90. They determined the adjacency spectra, Laplacian spectra, and signless Laplacian spectra of both constructions in terms of the corresponding spectra of $G_{1}$91 and $G_{1}$92. These formulas yield infinitely many pairs of cospectral graphs, and the Laplacian formulas produce new expander families from known ones. In particular, if $G_{1}$93 is an $G_{1}$94-expander of even degree $G_{1}$95, and $G_{1}$96 is the edgeless graph on $G_{1}$97 vertices, then $G_{1}$98 is again $G_{1}$99-regular and

$|V(G_{1})|$00

so the algebraic connectivity remains bounded below by a positive function of $|V(G_{1})|$01 (Liu et al., 2012).

The $|V(G_{1})|$02-graph $|V(G_{1})|$03 has vertex set $|V(G_{1})|$04, where $|V(G_{1})|$05 corresponds to the edges of $|V(G_{1})|$06, and two vertices of $|V(G_{1})|$07 are adjacent if and only if the corresponding elements of $|V(G_{1})|$08 are adjacent or incident. Using this graph, recent work defined the $|V(G_{1})|$09-vertex neighbourhood corona $|V(G_{1})|$10 and the $|V(G_{1})|$11-edge neighbourhood corona $|V(G_{1})|$12. For a connected regular $|V(G_{1})|$13 and an arbitrary regular $|V(G_{1})|$14, the adjacency and Laplacian spectra of these constructions are determined from the eigenvalues of the factors, again by Schur complements and a coronal method. As in earlier corona constructions, the resulting factorisations give non-regular $|V(G_{1})|$15-cospectral and $|V(G_{1})|$16-cospectral graphs (Mukherjee et al., 15 Sep 2025).

A further extension concerns normalized Laplacian spectra. The operations $|V(G_{1})|$17 and $|V(G_{1})|$18 attach copies of $|V(G_{1})|$19 and $|V(G_{1})|$20 simultaneously to the original and inserted vertices of $|V(G_{1})|$21. For connected regular $|V(G_{1})|$22, their normalized Laplacian spectra are determined in terms of the normalized Laplacian spectra of the factors. The same framework yields the number of spanning trees, the multiplicative degree-Kirchhoff index, and Kemeny’s constant, and it generalizes earlier subdivision-neighbourhood corona results (Wen et al., 2018).

The common pattern across these variants is explicit: neighbourhood attachment enlarges the graph in a highly structured way, and that structure survives in the matrix algebra. As a result, neighbourhood coronae form a family of graph products for which spectral data, automorphism data, and several derived invariants can be computed in closed or quasi-closed form.