Constructing signed strongly regular graphs via star complement technique
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Abstract
We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the wellknown method known as star complement technique.
Keywords
Signed graphs Eigenvalues Signed strongly regular graphsMathematics Subject Classification
05C50Introduction
 For any two adjacent vertices u, v, the following hold,$$\begin{aligned} \sum _{z\in N(u)\cap N(v)}s(zu)s(zv)=\lambda ^ss(uv). \end{aligned}$$
 For any two nonadjacent vertices u, v,$$\begin{aligned} \sum _{z\in N(u)\cap N(v)}s(zu)s(zv)=\mu ^s,\mu ^s. \end{aligned}$$
Preliminaries
Theorem A
Suppose the signed graph \((G,\Sigma )\) has just two distinct eigenvalues, then the graph G is a signed strongly regular graph with \(\mu ^s=0\).
Proof
The entries in diagonal of the matrix \((A^s)^2\) are the vertex degrees of G. On the other hand if \(\alpha ,\beta \) are the two distinct eigenvalues of \((G,\Sigma )\), then \(A^s\) provides the equality \((A^s)^2(\alpha +\beta ) A^s+\alpha \beta I=O\). The matrix \(A^s\) has zero diagonal; hence, the main diagonal entries of \((A^s)^2\) (and therefore the vertex degrees of G) are equal to \(\alpha \beta \), and this implies the regularity of G. Any off diagonal entry in place (i, j) of \((A^s)^2\) equals \(\sum\nolimits_{k=1}^{n} A^s(i,k)A^s(k,j)\), by the ordinary matrix multiplication rule. Hence, it equals \(\sum\nolimits_{z\in N(u)\cap N(v)}s(zu)s(zv)=\mu ^s.\) But from the above quadratic equation for \(A^s\), the entry (i, j) of \((A^s)^2\) must be equal to \((\alpha +\beta )A^s_{ij}\); therefore, the summation is zero if the vertices \(v_i\) and \(v_j\) are not adjacent and \((\alpha +\beta )s(v_iv_j)\) if they are adjacent. Therefore, \((G,\Sigma )\) is an SRG\(^s(n,k,(\alpha +\beta ),0\)) as desired.\(\square \)
In [3], the author has introduced several families of signed graphs with only two distinct eigenvalues. As an example, signed complete graphs with only two distinct eigenvalues have been mentioned. For more details about two graphs, see [2]. As an other family of signed strongly regular graphs, which is mentioned in [4], the line graph of the complete graph, say \(L(K_n)\), has been considered. We recall the construction from [4]. Let the vertices of the complete graph \(K_n\) are labeled with \(\{1,2,\ldots ,n\}\). It is known that the graph \(L(K_n)\) is an \({\text{SRG}}\left({n\atopwithdelims ()2},2n4,n2,4\right)\). The vertices of the graph \(L(K_n)\) are the \(n\atopwithdelims ()2\) two element sets \(\{a,b\}\), where \(a<b\), considering the following signature \(\Sigma \) on the graph \(L(K_n)\), which actually make it a signed strongly regular graph with parameters \({\text{SRG}}^s\left({n\atopwithdelims ()2},2n4,n3,0 \right)\).

Positive if \(a=c\) or \(b=d\).

Negative if \(a=d\) or \(b=c\).
Main result
Here using the wellknown method, i.e., star complement technique, we construct signed strongly regular graphs on 6 and 8 vertices; then, by the Kronecker product of matrices, we introduce some signed strongly regular graphs of larger order. We recall the star complement technique from [5]. Let A be an \(n\times n\) matrix, with an eigenvalue \(\mu \) of multiplicity k. A ksubset X of \({1,2,\ldots ,n}\) is called a star set if the matrix obtained from A by removing rows and columns corresponding to X does not have \(\mu \) as an eigenvalue. In graph theory context, a star set for an eigenvalue \(\mu \) in G is a subset X of vertices such that \(\mu \) is not an eigenvalue of \(G{\setminus} X\). The graph \(G{\setminus} X\) is called a star complement for \(\mu \) in G.
Theorem 1
Corollary 1
Note that the above results hold for any symmetric matrices, and specially, it applies to the signed graphs. For more convenient for two t vectors \({\mathbf{b}}_1,{\mathbf{b}}_2\), we denote \({\mathbf{b}}_1^T(\mu IC)^{1}{\mathbf{b}}_2\) by \(\langle {\mathbf{b}}_1,{\mathbf{b}}_2\rangle \) which is an inner product on the set of t vectors.
Corollary 2
Let H be a signed graph on t vertices and C be the adjacency of H, and we aim to construct a maximal signed graph that contains H as a star complement for the eigenvalue \(\mu \). We define the compatibility graph of H and \(\mu \) as follows: The vertices are \((0,\pm\,1)\) vectors \({\mathbf{b}}_i\) of dimension t such that \( \langle {\mathbf{b}}_i, {\mathbf{b}}_i \rangle =\mu \), and two vertices \({\mathbf{b}}_i\) and \({\mathbf{b}}_j\) are adjacent if and only if \(\langle {\mathbf{b}}_i, {\mathbf{b}}_j \rangle =0,\pm\,1.\) It is not hard to prove that the compatibility graph is finite (see [5]). Now the problem of finding maximal signed graphs having H as a star complement for eigenvalue \(\mu \) is equivalent to finding the maximal cliques in the compatibility graph of H and \(\mu \).
Examples
In this part, we present some examples to illustrate the method.
Example 1
 Case 1 The vertex u is in the same part of \(K_{2,2,2}\) where the vertex of degree two in H belongs. In this case, \(b_2=0,\) while \(b_1,b_3\) are \(\pm\,1\). Hence, the above equation becomesSince \(b_1^2+b_3^2=2\), hence \(b_1.b_3=1\), therefore one of the followings hold,$$\begin{aligned} \frac{3}{4}(b_1^2+b_3^2)+\frac{1}{2}b_1b_3=2. \end{aligned}$$$$\begin{aligned} b_1=1,b_3=1\quad {\text{or}}\quad b_1=\,1,b_3=\,1{.} \end{aligned}$$
 Case 2 In this case, u belongs to the third part of the graph \(\sum \) which is apart from the vertices of H. In this case, we have \(b_1^2+b_2^2+b_3^2=3\). Hence,Hence, \(b_1=b_3\) and \(b_2=1\) or \(\,1\). Thus, vectors which hold on the required equalities are shown in the following table. Note we have found the vectors via MATLAB programming.$$\begin{aligned}&\frac{3}{4}.3+b_1b_2+b_2b_3+\frac{1}{4}(2b_1b_3+1)=2\\&\Rightarrow 2(b_1b_2+b_2b_3)+b_1b_3=\,1 \end{aligned}$$
\(b_1\)
0
0
1
1
− 1
− 1
\( b_2\)
1
− 1
1
− 1
1
− 1
\(b_3\)
1
− 1
− 1
− 1
1
1
Example 2
En  Ve  

\({\mathbf{b}}_1\)  \({\mathbf{b}}_2\)  \({\mathbf{b}}_3 \)  \({\mathbf{b}}_4\)  \({\mathbf{b}}_5\)  \({\mathbf{b}}_6\)  \({\mathbf{b}}_7\)  \({\mathbf{b}}_8\)  \({\mathbf{b}}_9\)  \({\mathbf{b}}_{10}\)  \({\mathbf{b}}_{11}\)  \({\mathbf{b}}_{12}\)  \({\mathbf{b}}_{13}\)  \({\mathbf{b}}_{14}\)  \({\mathbf{b}}_{15}\)  \({\mathbf{b}}_{16}\)  
\(b_1\)  0  0  0  0  0  0  − 1  1  1  1  1  1  − 1  − 1  − 1  − 1 
\(b_2\)  1  1  − 1  − 1  − 1  1  1  − 1  1  0  − 1  − 1  1  1  − 1  0 
\( b_3 \)  1  − 1  1  − 1  1  − 1  1  − 1  − 1  0  1  − 1  1  − 1  1  0 
\(b_4\)  − 1  1  1  1  − 1  − 1  1  − 1  − 1  0  − 1  1  − 1  1  1  0 
The above example can be used to construct infinitely many complete tripartite signed graphs having only two distinct eigenvalues, or equivalently a signed strongly regular graph with \(\mu ^s=0\).
Theorem B
Let \(\sum \) be a signed graph with only two eigenvalues which are opposite, say \(\pm\,\lambda \), then the Kronecker product of the matrix \(H_2=\left( \begin{array}{cc} 1 &{}1 \\ 1 &{}\,1 \\ \end{array}\right) \) by the adjacency matrix of \(\sum \) gives the adjacency matrix of a signed graph with only two distinct eigenvalues \(\pm\,\sqrt{2}\lambda \).
Conclusion
Repeatedly applying the Kronecker product of the matrix \(H_2=\left( \begin{array}{cc} 1 &{}1 \\ 1 &{}\,1 \\ \end{array} \right) \) by the adjacency matrix of \(\sum \), we find infinitely many complete tripartite signed strongly regular graphs, whereas the same operation on the signed graph \(\Gamma \) will give infinitely many bipartite signed strongly regular graphs; the method can be used for several other examples to construct new families of signed graphs with the desired property.
Notes
References
 1.Akbari, S., Haemers, W.H., Maimani, H.R., Parsaei Majd, L.: Signed graphs cospectral with the path. arXiv:1709.09853v1Google Scholar
 2.Brouwer, A.E., Cohen, A.M., Neumaier, A.: DistanceRegular Graphs. Springer, Berlin (1989)CrossRefGoogle Scholar
 3.Ramezani, F.: On the signed graphs with two distinct eigenvalues (to appear in Utilitas Mathematica) Google Scholar
 4.Ramezani, F.: Signed strongly regular graphs. In: Proceeding of 48th Annual Iranian Mathematical Conference (2017)Google Scholar
 5.Tayfeh Rezaie, B.: Lecture Notes on the Star Complement Technique (Manuscript) Google Scholar
 6.Zaslavsky, T.: Signed graphs. Discrete Appl. Math. 4, 47–74 (1982)MathSciNetCrossRefGoogle Scholar
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