--- title: Neighbourhood Corona in Spectral Graph Theory url: https://www.emergentmind.com/topics/neighbourhood-corona type: topic --- # Neighbourhood Corona in Spectral Graph Theory 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 [1209.5906] [1606.03751] [2305.02964] [1212.0851]. ## 1. Definition and local structure Let \(G_{1}=(V(G_{1}),E(G_{1}))\) be a graph on \(n_{1}\) vertices \(v_{1},\dots,v_{n_{1}}\), and let \(G_{2}=(V(G_{2}),E(G_{2}))\) be a graph on \(n_{2}\) vertices. For each \(i=1,\dots,n_{1}\), one forms the \(i\)th copy of \(G_{2}\), and then joins every neighbour of \(v_{i}\) in \(G_{1}\) to every vertex in that \(i\)th copy. The resulting degree formulas are \[ \deg_{G_{1}\star G_{2}}(v_{i})=(n_{2}+1)\deg_{G_{1}}(v_{i}),\qquad \deg_{G_{1}\star G_{2}}(u_{j}^{i})=\deg_{G_{2}}(u_{j})+\deg_{G_{1}}(v_{i}), \] which already show the asymmetry between the original \(G_{1}\)-vertices and the vertices lying in the copies of \(G_{2}\) [1209.5906]. 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 \(S=(G,\sigma)\), where \(\sigma\) maps edges to \(\{\pm1\}\), and the signed \(s\)-neighbourhood corona \(S_{1}\star_{s}S_{2}\) is obtained by joining every neighbour of the \(i\)th vertex of \(S_{1}\) to every vertex in the \(i\)th copy of \(S_{2}\) with the same sign as the corresponding incident edge in \(S_{1}\). In subdivision-based variants, the operation is performed after replacing each edge of \(G_{1}\) by a two-edge path. In \(T\)-graph variants, the base graph is \(T(G_{1})\), whose vertices are the vertices and edges of \(G_{1}\) [2305.02964] [1212.0851] [2509.12342]. | Variant | Notation | Construction cue | |---|---|---| | Neighbourhood corona | \(G_{1}\star G_{2}\) | One copy of \(G_{1}\), \(n_{1}\) copies of \(G_{2}\), neighbours of \(v_i\) joined to the \(i\)th copy | | Signed \(s\)-neighbourhood corona | \(S_{1}\star_{s}S_{2}\) | Same pattern, with new edges carrying the sign of the relevant incident edge | | Subdivision-vertex neighbourhood corona | \(G_{1}\boxdot G_{2}\) | Built from \(\mathcal S(G_{1})\) and \(n_{1}\) copies of \(G_{2}\) | | Subdivision-edge neighbourhood corona | \(G_{1}\boxminus G_{2}\) | Built from \(\mathcal S(G_{1})\) and \(|I(G_{1})|\) copies of \(G_{2}\) | | \(T\)-vertex / \(T\)-edge neighbourhood corona | \(G_{1}\boxdot_{t}G_{2}\), \(G_{1}\boxminus_{t}G_{2}\) | Built from \(T(G_{1})\), 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 \(V(G_{1})\) and then the \(n_{1}\) copies of \(G_{2}\), the adjacency matrix is \[ A(G_{1}\star G_{2})= \begin{pmatrix} A(G_{1}) & \mathbf{1}_{n_{2}}^{T}\otimes A(G_{1})\\[6pt] (\mathbf{1}_{n_{2}}^{T}\otimes A(G_{1}))^{T} & I_{n_{1}}\otimes A(G_{2}) \end{pmatrix}. \] The key auxiliary object is the coronal \[ \Gamma_{M}(x)=\mathbf{1}^{T}(xI-M)^{-1}\mathbf{1}, \] 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 \(S_{1}\star_{s}S_{2}\), with the off-diagonal blocks determined by the signed adjacency of \(S_{1}\) [1209.5906] [2305.02964]. Subdivision-based neighbourhood coronae replace the adjacency blocks involving \(A(G_{1})\) by incidence-matrix blocks. For example, in the subdivision-vertex neighbourhood corona \(G_{1}\boxdot G_{2}\), the matrix formulas are written in terms of the vertex-edge incidence matrix \(R\) of \(G_{1}\). The same pattern persists in normalized Laplacian and \(T\)-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 [1212.0851] [1806.10133] [2509.12342]. ## 3. Spectra of the ordinary neighbourhood corona The basic adjacency theorem of Liu and Zhou expresses the characteristic polynomial of \(G_{1}\star G_{2}\) directly in terms of the adjacency spectrum of \(G_{1}\) and the \(A\)-coronal of \(G_{2}\): \[ \phi\bigl(A(G_{1}\star G_{2});x\bigr) = \bigl(\phi(A(G_{2});x)\bigr)^{n_{1}} \prod_{i=1}^{n_{1}} \bigl(x-\lambda_{i}(G_{1})-\Gamma_{A(G_{2})}(x)\lambda_{i}(G_{1})^{2}\bigr). \] Here \(\lambda_{1}(G_{1}),\dots,\lambda_{n_{1}}(G_{1})\) are the eigenvalues of \(A(G_{1})\). When \(G_{1}\) is \(r_{1}\)-regular, parallel formulas hold for the signless Laplacian and Laplacian characteristic polynomials, with \(\Gamma_{Q(G_{2})}(x-r_{1})\) and \(\Gamma_{L(G_{2})}(x-r_{1})\) replacing the adjacency coronal. Since each row-sum of \(L(G_{2})\) is \(0\), one has \(\Gamma_{L(G_{2})}(y)=n_{2}/y\), which yields explicit Laplacian eigenvalue formulas [1209.5906]. These factorisations have two immediate uses. First, they produce cospectral constructions. If \(G\) and \(G'\) are \(A\)-cospectral and \(H\) is any graph, then \(G\star H\) and \(G'\star H\) are \(A\)-cospectral. Likewise, if \(G_{1},G_{2}\) are \(Q\)-cospectral regular graphs, then \(G_{1}\star H\) and \(G_{2}\star H\) are \(Q\)-cospectral. Second, the Laplacian formulas support expander constructions. If \(\{G_{i}\}\) is a family of non-complete \(k\)-regular \(\varepsilon\)-expanders, and \(H\neq K_{n}\) is a fixed connected \(r\)-regular graph with \(nk\) even, then after adding edges so as to make each \(G_{i}\star H\) \((n+1)k\)-regular, the resulting family \(\{G_{i}(H)\}\) is a family of \(\delta(n,k,\varepsilon)\)-expanders [1209.5906]. ## 4. Signed neighbourhood coronae In signed graph theory, two different neighbourhood-corona constructions have been studied. The first is the signed \(s\)-neighbourhood corona \(S_{1}\star_{s}S_{2}\), where \(S=(G,\sigma)\) is a signed graph and the new edges inherit the sign of the corresponding incident edge of \(S_{1}\). If \(S_{2}\) is \(r_{2}\)-net-regular with adjacency eigenvalues \(\sigma_{A}(S_{2})=\{\theta_{1}(S_{2}),\dots,\theta_{n_{2}}(S_{2})\}\), with \(\theta_{n_{2}}(S_{2})=r_{2}\), and \(\sigma_{A}(S_{1})=\{\theta_{1}(S_{1}),\dots,\theta_{n_{1}}(S_{1})\}\), then the adjacency spectrum of \(S_{1}\star_{s}S_{2}\) consists of: each \(\theta_{j}(S_{2})\), \(j\neq n_{2}\), repeated \(n_{1}\) times, and for each \(i=1,\dots,n_{1}\) two new eigenvalues \[ \lambda_{i}^{\pm} =\frac{\theta_{i}(S_{1})+r_{2} \pm\sqrt{(r_{2}-\theta_{i}(S_{1}))^{2}+4n_{2}\theta_{i}(S_{1})^{2}}}{2}. \] The same paper determines the Laplacian spectrum for regular \(S_{1}\) and regular and net-regular \(S_{2}\), and the net Laplacian spectrum for net-regular \(S_{1}\) and arbitrary \(S_{2}\). It also identifies a “copy-eigenvalue” phenomenon: each eigenvalue of the second factor, except a distinguished pole, appears with multiplicity \(n_{1}\), while the remaining \(2n_{1}\) eigenvalues arise from quadratic or rational equations. As consequences, the authors obtain signed graphs with \(4\) and \(5\) 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 [2305.02964]. A second signed model, denoted \(\Gamma_{1}*\Gamma_{2}\), uses canonical vertex markings. If \(\Gamma_{1}=(H_{1},\sigma_{1},\mu_{1})\) and \(\Gamma_{2}=(H_{2},\sigma_{2},\mu_{2})\), then each new edge joining a neighbour \(u\) of \(u_{i}\) in \(\Gamma_{1}\) to a vertex \(v_{j}^{i}\) in the \(i\)th copy of \(\Gamma_{2}\) is given sign \[ \sigma_{\mathrm{new}}(u,v_{j}^{i})=\mu_{1}(u)\cdot \mu_{2}(v_{j}). \] Its adjacency characteristic polynomial is \[ f(A(\Gamma_{1}*\Gamma_{2});x) = [f(A(\Gamma_{2});x)]^{n_{1}} \prod_{i=1}^{n_{1}} \bigl(x-\lambda_{i}(\Gamma_{1})-\chi_{A(\Gamma_{2})}(x)\lambda_{i}(\Gamma_{1})^{2}\bigr), \] where \(\chi_{A(\Gamma)}(x)=\mu^{T}(xI-A(\Gamma))^{-1}\mu\). This construction also yields Laplacian and signless Laplacian formulas when \(\Gamma_{1}\) is regular. A notable structural result is that balanced factors do not by themselves guarantee a balanced product: even when both \(\Gamma_{1}\) and \(\Gamma_{2}\) are balanced, \(\Gamma_{1}*\Gamma_{2}\) fails to be balanced if and only if one of the factors contains one of three specified “bad” edge types [2310.12814]. ## 5. Automorphisms, distinguishing invariants, and coloring The automorphism group of \(G_{1}\star G_{2}\) is tightly controlled by the structure of the base graph. If \(G_{1}\neq K_{1}\) is connected and \(f\in\mathrm{Aut}(G_{1}\star G_{2})\), then \(f\) restricts to an automorphism of \(G_{1}\), and the \(n_{1}\) copies of \(G_{2}\) are permuted among themselves. More precisely, every automorphism is uniquely determined by a pair \[ \bigl(g\in\mathrm{Aut}(G_{1}),\;h_{1},\dots,h_{n_{1}}\in\mathrm{Aut}(G_{2})\bigr), \] where \(g\) acts on the central copy of \(G_{1}\) and each \(h_{i}\) acts within the \(i\)th copy of \(G_{2}\) before that copy is moved according to \(g\) [1606.03751]. This automorphism description supports upper bounds on symmetry-breaking parameters. For connected \(G\) of order \(n>1\), \[ D(G\star K_{1})\le D(G),\qquad D'(G\star K_{1})\le D'(G). \] More generally, if \(D_{1}=D(G_{1})\), \(D_{2}=D(G_{2})\), \(y_{0}=1\), \(y_{1}=D_{2}\), and for \(m\ge2\), \[ y_{m}=D_{2}+\sum_{i=1}^{m-1}\binom{m-1}{i}\binom{D_{2}}{i+1}, \qquad M=\min\Bigl\{k:\sum_{m=0}^{k}y_{m}\ge D_{1}\Bigr\}, \] then \[ D(G_{1}\star G_{2})\le \max\{D_{1},D_{2}+M\}. \] For the distinguishing index, \[ D'(G_{1}\star G_{2})\le \max\{D'(G_{1}),D'(G_{2})\} \] for connected \(G_{1},G_{2}\) with \(|V(G_{1})|>1\) [1606.03751]. Coloring theory enters through subdivision-based neighbourhood coronae. For graphs \(G\) and \(H\) that are paths, cycles, or stars, the subdivision-vertex neighbourhood corona \(G\boxdot H\) has explicitly determined \(b\)-chromatic number, and the same is true for \(G\boxdot K_{n}\). The paper also establishes the \(b\)-chromatic number for graphs \(K_{n}\boxdot G\) having \(m\)-degree not greater than \(n+2\). Representative formulas include \[ \varphi(S_{n}\boxdot K_{t})=\min\{n,t+2\}+t \] and \[ \varphi(S_{n}\boxdot S_{t})= \begin{cases} 2n+1,&n\le (t+1)/2,\\ t+2,&(t+1)/2