New constructions of non-regular cospectral graphs

We consider two types of joins of graphs $G_{1}$ and $G_{2}$, $G_{1}\veebar G_{2}$ - the Neighbors Splitting Join and $G_{1}\underset{=}{\lor}G_{2}$ - the Non Neighbors Splitting Join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of these joins. When $G_{1}$ and $G_{2}$ are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of $G_{1}\underset{=}{\lor}G_{2}$ and the normalized Laplacian spectrum of $G_{1}\veebar G_{2}$ and $G_{1}\underset{=}{\lor}G_{2}$. We use these results to construct non regular, non isomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian , signless Laplacian and normalized Laplacian.


Introduction
Spectral graph theory is the study of graphs via the spectrum of matrices associated with them [3,6,8,22,27].
The graphs in this paper are undirected and simple.There are several matrices associated with a graph and we consider four of them; the adjacency matrix, the Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix.
Let G = (V (G) , E (G)) be a graph with vertex set V (G) = {v 1 , v 2 , ..., v n } and edge set E (G).(with the convention that if the degree of vertex v i in G is 0, then (d i ) − 1 2 = 0).In other words, , if i ̸ = j and v i is adjacent to v j ; 0, otherwise.
Notation 2.4.For an n × n matrix M , we denote the characteristic polynomial det (xI n − M ) of M by f M (x), where I n is the identity matrix of order n.In particular, for a graph G, f X(G) (x) is the X-characteristic polynomial of G, for X ∈ {A, L, Q, L}.The roots of the X-characteristic polynomial of G are the X-eigenvalues of G and the collection of the X-eigenvalues, including multiplicities, is called the X-spectrum of G.
Notation 2.5.The multiplicity of an eigenvalue λ is denoted by a superscript above λ.
Remark 2.7.If are the eigenvalues of A (G), L (G), Q (G) and L (G), respectively.Then Remark 2.8.If G is a r-regular graph, then µ i (G) = r − λ i (G), ν i (G) = r + λ i (G) and δ i (G) = 1 − 1 r λ (G).Definition 2.9.Two graphs G and H are X-cospectral if they have the same X-spectrum.If X-cospectral graphs are not isomorphic we say that they are XNICS.Definition 2.10.Let S be a subset of {A, L, Q, L} .The graphs G and H are SNICS if they are XNICS for all X ∈ S. Definition 2.11.A graph G is determined by its X-spectrum if every graph H that is X-cospectral with G is isomorphic to G.
A basic probem in spectral graph theory, [28,29], is determining which graphs are determined by their spectrum or finding non isomorphic X-cospectral graphs.
Theorem 2.12.( [28]) If G is regular , then the following are equivalent; • G is determined by its A-spectrum, • G is determined by its L-spectrum, • G is determined by its Q-spectrum, • G is determined by its L-spectrum.
Thus, for regular graphs G and H, we say that G and H are cospectral if they are X-cospectral with respect to any X ∈ {A, L, Q, L} .Proposition 2.13.( [28]) Every regular graph with less than ten vertices is determined by its spectrum.
Example 2.14.The following graphs are regular and cospectral.They are non isomorphic since in G there is an edge that lies in three triangles but there is no such edge in H.In recent years, several researchers studied the spectral properties of graphs which are constructed by graph operations.These operations include disjoint union, the Cartesian product, the Kronocker product, the strong product, the lexicographic product, the rooted product, the corona, the edge corona, the neighbourhood corona etc.We refer the reader to [1,2,8,13,16,21,9,12,24,23,25,26] and the references therein for more graph operations and the results on the spectra of these graphs.
Many operations are based on the join of graphs.
Definition 2.15.( [14]) The join of two graphs is their disjoint union together with all the edges that connect all the vertices of the first graph with all the vertices of the second graph.
Butler [4]  Such examples can be constructed using special join operation defined by Lu, Ma and Zhang [19] and a variant of this operation, suggested in this paper.
We refer to the splitting V -vertex join as NS (Neighbors Splitting) join and define a new type of join, NNS (Non Neighbors Splitting) join.
Definition 2.18.Let G 1 and G 2 be two vertex disjoint graphs with  • 1 n denotes n×1 column whose all entries are 1, • O s×t denotes the zero matrix of order s × t, • adj (A) denotes the adjugate of A.
• G denotes the complement of graph G.
Definition 3.2.( [7,20]) The coronal Γ M (x) of an n×n matrix M is the sum of the entries of the inverse of the characteristic matrix of M , that is, 7,20]) Let M be an n×n matrix with all row sums equal to r ( for example, the adjacency matrix of a r-regular graph).Then Definition 3.4.Let M be a block matrix such that its blocks A and D are square.If A is invertible, the Schur complement of Issai Schur proved the following lemma.Lemma 3.6.Let M be a block matrix where A, B, and C are square matrices of order n 1 and D is a square matrix of order n 2 .Then the Schur Proof.The charactristic matrix of M is .
The Schur complement of (xI n2 − D) is Lemma 3.7.( [8]).If A is an n×n real matrix and α is an real number, then 4 The characteristic polynomials of the NNS join and the NS join In [19] the authers compute the adjacency, Laplacian and signless Laplacian characteristic polynomial of G 1 ⊻G 2 where G 1 and G 2 are regular.
Here we compute the characteristic polynomials of where G 1 and G 2 are arbitrary graphs.The proofs for the two joins (NS and NNS) are quite similar and use Lemma 3.5 (twice), Lemma 3.6 and Lemma 3.7.The results are used to construct non regular {A, L, Q}NICS graphs.

Adjacency characteristic polynomial
Theorem 4.1.Let G i be a graph on n i vertices for i = 1, 2. Then (a) ]︂ .
Proof.We prove (a).The proof of (b) is similar.
With a suitable ordering of the vertices of by the Lemma of Schur (Lemma 3.5).

Laplacian characteristic polynomial
In this section, we derive the Laplacian characteristic polynomials of The Laplacian characteristic polynomial is by the Lemma of Schur (Lemma 3.5).
By Lemma 3.6, Using again Lemma 3.5 , we get By Lemma 3.7 we get The proof of (b) is similar.

Signless Laplacian characteristic polynomial
Theorem 4.3.Let G i be a graph on n i vertices for i = 1, 2. Then (a) ]︂ .
Proof.The proof is similar to the proof of Theorem 4.2.
since the matrices A(F ) and A(H) have the same coronal (Lemma 3.3) and the same characteristic polynomial.This completes the proof of (a).
The proof of (b) is similar.
Corollary 4.5.Let F and H be r-regular non isomorphic cospectral graphs.Then for every G, Proof.The proof is similar to the proof of Corollary 4.4.
The following examples demonstrate the importance of the regularity of the graphs F and H.

2(2r1+n2)
for i = 2, 3, ..., n 1 ; for i = 2, 3, ..., n 1 ; • the three roots of the equation Proof.Let u 1 , u 2 , . . ., u n1 be the vertices of G 1 , u ′ 1 , u ′ 2 , . . ., u ′ n1 be the vertices added by the splitting and v 1 , v 2 , ..., v n2 be the vertices of G 2 .Under this vertex partitioning the adjacency matrix of By simple calculation we get We prove the theorem by constructing an orthogonal basis of eigenvectors of For i = 2, ..., n 1 , let X i be an eigenvector of A(G 1 ) corresponding to the eigenvalue λ i (G 1 ).We now look for a non zero real number α such that (︂ so, α must be a root of the equation . Substituting the values of α in the right side of (5.1), we get by Remark 2.8 that So far, we obtained To find three additional eigenvalues, we look for eigenvectors of Notice that α ̸ = 0, since if α = 0 then α = β = γ = 0 and also x ̸ = 1 since x = 1 implies that α = 0.
Dividing by α, we get the following cubic equation and this completes the proof.
Now   6 Spectra of NNS Joins In this section we compute the A-spectrum, L-spectrum, Q-spectrum and L-spectrum of G 1 ∨ = G 2 where G 1 and G 2 are regular.We use it to answer Question 2.16 by constructing pairs of non regular {A, L, Q, L}NICS graphs.

A-spectra of NNS join
The adjacency matrix of G 1 ∨ = G 2 can be written in a block form Theorem 6.1.Let G 1 be a r 1 -regular graph with n 1 vertices and G 2 be a r 2 -regular graph with n 2 vertices.
Then the adjacency spectrum of G 1 ∨ = G 2 consists of: for each j = 2, 3, . . ., n 2 ; • (ii) two roots of the equation )︁ = 0 for each i = 2, 3, . . ., n 1 ; • (iii) the three roots of the equation Proof.By Theorem 4.1, the adjacency characteristic polynomial of Since G 1 and G 2 are regular, we can use Lemma 3.3 to get Thus, based on Definition 3.2 and Lemma 3.3, we have

L-spectra of NNS join
The degrees of the vertices of Theorem 6.2.Let G 1 be a r 1 -regular graph with n 1 vertices and G 2 be a r 2 -regular graph with n 2 vertices.Then the Laplacian spectrum of G 1 ∨ = G 2 consists of: for each j = 2, 3, ..., n 2 ; • two roots of the equation for each i = 2, 3, ..., n 1 ; • the three roots of the equation ]︃ .

Q-spectra of NNS join
• two roots of the equation • the three roots of the equation Proof.The proof is similar to the proof of Theorem 6.2.

L-spectra of NNS join
Let G 1 be a r 1 -regular graph on order n 1 .Let S be a subset of {2, 3, . . ., n 1 } such that δ i (G 1 ) = 1 + 1 r1 for i ∈ S, and denote the cardinality of S by n(S).Let G 2 be a r 2 -regular graph on order n 2 .In the following theorem we determine the normalized Laplacian spectrum of G 1 ∨ = G 2 in terms of the normalized Laplacian eigenvalues of G 1 and G 2 .The proof is slightly more complicated than the proof of Theorem 5.1 and we consider three cases.Theorem 6.4.
For i = 2, ..., n 1 , let X i be an eigenvector of A(G 1 ) corresponding to the eigenvalue λ This completes the proof of (b).

Definition 2 . 1 .⎩ 1 ,
The adjacency matrix of G, A (G) , is defined by;(A(G)) ij = ⎧ ⎨ if v i and v j are adjacent; 0, otherwise.Let d i = d G (v i ) be the degree of vertex v i in G,and let D (G) be the diagonal matrix with diagonal entries d 1 , d 2 , ..., d n .Definition 2.2.The Laplacian matrix, L (G) , and the signless Laplacian matrix, Q (G) , of G are defined as L (G) = D (G) − A (G) and Q (G) = D (G) + A (G).

Example 2 . 19 .
Let G 1 and G 2 be the path P 4 and the path P 2 , respectively.The graphs P 4 ∨ = P 2 and P 4 ⊻ P 2 are given in Figure2.2.

Corollary 4 . 4 .=F
Let F and H be r-regular non isomorphic cospectral graphs.Then for every G, a) G ⊻ F and G ⊻ H are {A, Q, L}NICS.b) G ∨ and G ∨ = H are {A, Q, L}NICS.Proof.(a) G ⊻ F and G ⊻ H are non isomorphic since F and H are non isomorphic.By Theorems 4.1, 4.2 and 4.3,

Example 5 . 3 .
Let G 1 = H 1 = C 4 , and choose G 2 = G and H 2 = H where G and H are graphs in Figure 2.1, then the graphs in Figure 2.1 are {A, L, Q, L}NICS.

Theorem 6 . 3 .=G 2
Let G 1 be a r 1 -regular graph with n 1 vertices and G 2 be a r 2 -regular graph with n 2 vertices.Then the signless Laplacian spectrum of G 1 ∨ consists of:

1 n1+n2− 1 )Corollary 6 . 5 .H 2
[n(S)] are eigenvalues of L the proof of (c).Now we can give another answer to Question 2.16 by constructing several pairs of non regular {A, L, Q, L}NICS graphs.Let G 1 , H 1 be cospectral regular graphs and G 2 , H 2 be non isomorphic, regular and cospectral graphs.Then G 1 ∨ = G 2 and H 1 ∨ = are non regular {A, L, Q, L}NICS.Proof.G 1 ∨ = G 2 and H 1 ∨ = H 2 are non isomorphic since G 2 and H 2 are non isomorphic.By Theorems 6.1, 6.2, 6.3 and 6.4 , we get that G 1 ∨ = G 2 and H 1 ∨ = H 2 are non regular {A, L, Q, L}NICS.Example 6.6.Let G 1 = H 1 = C 4 , and choose G 2 = G and H 2 = H where G and H are graphs in Figure 2.1, then the graphs in Figure 6.1 are {A, L, Q, L}NICS.
5 L − Spectra of NS Joins Let G i , H i be r i -regular graphs, i = 1, 2. Lu, Ma and Zhang showed that if G 1 and H 1 are cospectral and G 2 and H 2 are cospectral and non isomorphic, then G 1 ⊻ G 2 and H 1 ⊻ H 2 are {A, L, Q}NICS.In this section,we extend this result by showing that G 1 ⊻ G 2 and H 1 ⊻ H 2 are {A, L, Q, L}NICS.To do it we determine the spectrum of the normalized Laplacian of the graph G 1 ⊻ G 2 .Theorem 5.1.Let G 1 be a r 1 -regular graph with n 1 vertices and G 2 be a r 2 -regular graph with n 2 vertices.Then the normalized Laplacian spectrum of G 1 ⊻ G 2 consists of: we can answer Question 2.16 by constructing pairs of non regular {A, L, Q, L}NICS graphs.Corollary 5.2.Let G i , H i be r i -regular graphs, i = 1, 2. If G 1 and H 1 are cospectral and G 2 and H 2 are cospectral and non isomorphic then G 1 ⊻ G 2 and H 1 ⊻ H 2 are {A, L, Q, L}NICS.
Proof.G 1 ⊻ G 2 and H 1 ⊻ H 2 are non isomorphic since G 2 and H 2 are non isomorphic.By Theorem 5.1 and Theorems 3.1, 3.2 and 3.3 in [19], the graphs H 1 ⊻ H 2 are {A, L, Q, L}NICS.