S. Pirzada^a , Mir Riyaz Ul Rashid^a , Amir Rehman^b , Edy Tri Baskoroc,d

pirzadasd@kashmiruniversity.ac.in, mirriyaz4097@gmail.com, aamirnajar786@gmail.com, ebaskoro@itb.ac.id

A signed graph Σ is the ordered pair (G, σ), where G = (V, E) is a finite simple graph, called the underlying graph, and σ : E(G) −→ {+1, −1} is a sign function or a signature of Σ. Let (Kn, σ) be a complete signed graph with n vertices. In this paper, we give a complete description of the adjacency, Laplacian and net Laplacian spectrum of a complete signed graph (Kn, σ) whenever its negative edges induce either a complete tripartite graph or a friendship graph. This is an addition to the class of complete signed graphs whose spectra is completely known.

Keywords: complete signed graph; complete tripartite graph; friendship graph; spectrum; Laplacian spectrum; net Laplacian spectrum

Mathematics Subject Classification: 05C22; 05C50

DOI: 10.5614/ejgta.2026.14.1.16

1. Introduction

Let G be a simple graph of order n with vertex set V (G) = {v1, v2, . . . , vn} and the edge set E(G). The signed graph Σ = (G, σ) is a graph G together with a function σ : E(G) −→ {+1, −1} called the signature of G. If σ(e) = 1 (respectively, σ(e) = −1) for every edge e, then σ is called the all-positive (respectively, all-negative) signature and Σ = (G, σ) is called an all-positive (respectively, all-negative) signed graph. The underlying graph G is interpreted as a signed graph where all its edges are positive. Recent work on signed graphs which can be useful in future studies can be seen in [3, 5, 19].

Received: 4 November 2025, Revised: 5 April 2026, Accepted: 12 April 2026.

^aDepartment of Mathematics, University of Kashmir, Srinagar, India

^bDepartment of Mathematics, Govt. Degree College, Frisal, J&K, India

^cCombinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia

^dCenter for Research Collaboration on Graph Theory and Combinatorics, Indonesia

The adjacency matrix A(Σ) = (aij ) of Σ is an n × n matrix such that aij = σ(ij) if i and j are adjacent, and 0 otherwise. The Laplacian matrix of Σ is defined as L(Σ) = D(Σ) − A(Σ), where D(Σ) is the diagonal matrix of vertex degrees in Σ. For a vertex v of Σ, the net degree of v is the difference between the number of positive edges and the number of negative edges incident with this vertex. Accordingly, D^±(Σ) is the diagonal matrix of net degrees and the net Laplacian matrix N(Σ) is defined as N(Σ) = D^±(Σ) − A(Σ).

We write ΦM(x) and Spec(M) to denote the characteristic polynomial and the spectrum of a square matrix M, respectively. In particular, if M is the adjacency matrix of a signed graph Σ, this notation is simplified to ΦΣ(x) and Spec(Σ). Also, we will write LSpec(Σ) and NSpec(Σ) for the Laplacian and net Laplacian spectrum, respectively. The spectrum of any matrix is considered as a multiset, say λ m¹ 1 , λ^m2 2 , . . . , λ^mt t , in which an exponent denotes the algebraic multiplicity of the corresponding eigenvalue.

In contrast to the former two matrices, the net Laplacian is less studied. It can be considered as a counterpart to the standard Laplacian matrix of an ordinary graph; moreover, in case of graphs these matrices coincide. Although it appears sporadically in earlier publications, it is known under the given name since 2020 [17]. Observe that 0 belongs to the net Laplacian spectrum of every signed graph; an associated eigenvector is the all-1 vector. Other spectral properties and relationships with the standard Laplacian can be found in [6, 15]. A relevance in control theory has been recognized in [4, 16]. Some spectral properties and relations with other graph matrices are given in [15, 18]. In [6], the net Laplacian spectrum of complete signed graphs is examined and signed graphs with exactly two distinct net Laplacian eigenvalues are characterized.

In recent years, the spectral analysis of signed graphs has attracted considerable attention from researchers. In [1], Akbari et al. determined the spectrum of complete signed graphs and complete bipartite signed graphs whenever negative edges form a matching. More recently, Pirzada et al. [8], determined the spectra of complete bipartite signed graphs whose negative edges induce either a matching, or a complete bipartite graph, or a double star graph. For similar results on signed graphs, we refer the reader to [2, 7, 11, 12, 16, 17]. Some recent work on spectra of signed graphs can be seen in [9, 10, 13, 14].

A complete tripartite graph is a graph whose vertex set is partitioned into three disjoint subsets, with every vertex in one subset adjacent to all vertices in the other two subsets and there are no edges within the same subset. It is denoted by Ka,b,c. A friendship graph is the graph obtained by joining p triangles at a single common vertex, so that each pair of triangles shares exactly this vertex. It is denoted by F2p+1.

The results in this paper are mainly inspired by the work presented in [1] and [2]. To present our results, we consider Σ as a complete signed graph. In the following sections, we determine the spectrum, Laplacian spectrum, and net Laplacian spectrum of Σ whenever its negative edges induce either a complete tripartite graph or a friendship graph.

Section 2 serves as a preliminary section, providing the necessary notation and an important lemma used throughout the paper. Our main results are presented in Sections 3 and 4: the former addresses the case where the negative edges form a complete tripartite graph, while the latter focuses on the case where they form a friendship graph. Finally, Section 5 presents a brief conclusion.

2. Preliminaries

We write \(I_p\) to denote the \(p \times p\) identity matrix, and \(J_{p \times q}\) (or \(J_p\) if p = q) for the \(p \times q\) all-1 matrix. Also, we denote the all-1 (resp. all-0) column vector of size p by \(\mathbf{j}_p\) (\(\mathbf{o}_p\)).

In what follows, we will frequently deal with a complete signed graph \(\Sigma = (K_n, \sigma)\) with n vertices. In this context, if H is a subgraph of \(K_n\), then we write \((K_n, H^-)\) to denote the complete signed graph underlined by \(K_n\) whose edge is negative if and only if it belongs to H.

We now consider a known result about the spectrum of a matrix. Let M be an \(n \times n\) matrix blocked as

\[M = \begin{pmatrix} M_{11} & M_{12} & \cdots & M_{1t} \\ M_{21} & M_{22} & \cdots & M_{2t} \\ \vdots & \vdots & \ddots & \vdots \\ M_{t1} & M_{t2} & \cdots & M_{tt} \end{pmatrix},\]

where \(M_{ij}\) is an \(n_i \times n_j\) matrix, for \(1 \le i, j \le t\), and \(n = \sum_{i=1}^t n_i\). If \(b_{ij}\) denotes the average row sum of \(M_{ij}\), then \(Q = (b_{ij})\) is called the quotient matrix of M. If, in addition, \(M_{ij}\) has a constant row sum, then Q is called the quotient matrix of M. The following result is well known in the literature.

Theorem 2.1 ([20, Theorem 2.3]). Let Q be the equitable quotient matrix of M. Then \(\operatorname{Spec}(Q) \subseteq \operatorname{Spec}(M)\).

3. Adjacency, Laplacian and Net Laplacian Spectrum of \((K_n, K_{a,b,c}^-)\)

We first deal with complete signed graphs \(\Sigma\) whose negative edges induce a complete tripartite graph. We first compute the spectrum.

Theorem 3.1. Let \(\Sigma \cong (K_n, K_{a,b,c}^-)\). The spectrum of \(\Sigma\) consists of If \(n \neq a + b + c\)

• (i) -1 with multiplicity n-4,
• (ii) the four roots of \(x^4 + (4-n)x^3 + (6-3n)x^2 + C_1x + C_0 = 0\), where \(C_1 = -4a^2b 4a^2c 4b^2a 8abc 4c^2a 4b^2c 4c^2b + 4abn + 4acn + 4bcn + 4 3n\) and \(C_0 = 1 n 4a^2b 4a^2c 4b^2a 8abc 4c^2a 4b^2c 4c^2b + 4abn + 4acn + 4bcn + 16bca^2 + 16acb^2 + 16abc^2 16abcn\)

If n = a + b + c

• (i) -1 with multiplicity n-3,
• (ii) the three roots of \(x^3 (n-3)x^2 + (3-2n)x n + 1 + 4abc = 0\).

Proof. Case 1. With a suitable labeling of the vertices of \(\Sigma\), its adjacency matrix is

\[A(\Sigma) = \begin{pmatrix} J_a - I_a & -J_{a \times b} & -J_{a \times c} & J_{a \times (n-a-b-c)} \\ -J_{b \times a} & J_b - I_b & -J_{b \times c} & J_{b \times (n-a-b-c)} \\ -J_{c \times a} & -J_{c \times b} & J_c - I_c & J_{c \times (n-a-b-c)} \\ J_{(n-a-b-c) \times a} & -J_{(n-a-b-c) \times b} & J_{(n-a-b-c) \times c} & J_{n-a-b-c} - I_{n-a-b-c} \end{pmatrix}.\]

Let \(\mathbf{x} \in \mathbb{R}^a\), \(\mathbf{y} \in \mathbb{R}^b\), \(\mathbf{z} \in \mathbb{R}^c\) and \(\mathbf{t} \in \mathbb{R}^{n-a-b-c}\) be column vectors orthogonal to \(\mathbf{j}_a\), \(\mathbf{j}_b\), \(\mathbf{j}_c\) and \(\mathbf{j}_{n-a-b-c}\) respectively. By setting \(\mathbf{w} = (\mathbf{x}^\intercal, \mathbf{y}^\intercal, \mathbf{z}^\intercal, \mathbf{t}^\intercal)^\intercal \in \mathbb{R}^n\), we immediately obtain

\[A(\Sigma)\mathbf{w} = (-\mathbf{x}^{\mathsf{T}}, -\mathbf{y}^{\mathsf{T}}, -\mathbf{z}^{\mathsf{T}}, -\mathbf{t}^{\mathsf{T}})^{\mathsf{T}} = -1\mathbf{w}.\]

It follows that w is an eigenvector associated with the eigenvalue -1. Since there are a-1+b-1+c-1+n-a-b-c-1(=n-4) such linearly independent column vectors of size n that are orthogonal to \(\mathbf{j}_n\), the algebraic multiplicity of -1 as an eigenvalue of \(\Sigma\) is (at least) n-4, which proves (i).

The remaining four eigenvalues are obtained from the \(4 \times 4\) equitable quotient matrix Q of \(A(\Sigma)\) according to Theorem 2.1. The matrix Q is given by

\[Q = \begin{pmatrix} a-1 & -b & -c & n-a-b-c \\ -a & b-1 & -c & n-a-b-c \\ -a & -b & c-1 & n-a-b-c \\ a & b & c & n-a-b-c-1 \end{pmatrix}.\]

The characteristic polynomial of Q is computed as follows. The trace of a matrix Q is given by

\[\operatorname{Tr}(Q) = n - 4\]

We calculate the \(2 \times 2\) principal minors of Q

\[M_{12} = \det \begin{pmatrix} a-1 & -b \\ -a & b-1 \end{pmatrix} = (a-1)(b-1) - ab = 1 - a - b.\] \[M_{13} = \det \begin{pmatrix} a-1 & -c \\ -a & c-1 \end{pmatrix} = (a-1)(c-1) - ac = 1 - a - c.\] \[M_{14} = \det \begin{pmatrix} a-1 & n-a-b-c \\ a & n-a-b-c-1 \end{pmatrix} = b+c+1-n.\] \[M_{23} = \det \begin{pmatrix} b-1 & -c \\ -b & c-1 \end{pmatrix} = (b-1)(c-1) + bc = 1 - b - c.\] \[M_{24} = \det \begin{pmatrix} b-1 & n-a-b-c \\ b & n-a-b-c-1 \end{pmatrix} = a+c+1-n.\] \[M_{34} = \det \begin{pmatrix} c-1 & n-a-b-c \\ a & n-a-b-c-1 \end{pmatrix} = b+a+1-n.\]

Clearly, the sum of all \(2 \times 2\) principal minors is \(M_2 = 6 - 3n\). Further, the \(3 \times 3\) principal minors of Q are given by

\[M'_{11} = \det \begin{pmatrix} b-1 & -c & n-a-b-c \\ -b & c-1 & n-a-b-c \\ b & c & n-a-b-c-1 \end{pmatrix} = a(4bc-1) + 4b^2c + 4bc(c-n) + n-1.\] \[M'_{22} = \det \begin{pmatrix} a-1 & -c & n-a-b-c \\ -a & c-1 & n-a-b-c \\ a & c & n-a-b-c-1 \end{pmatrix} = 4c^2c + b(4ac-1) + 4ac(c-n) + n-1.\] \[M'_{33} = \det \begin{pmatrix} a-1 & -b & n-a-b-c \\ -a & b-1 & n-a-b-c \\ a & b & n-a-b-c-1 \end{pmatrix} = 4a^2b + 4ab(b+c-n) - c+n-1.\]

Computation of the eigenvalues of complete signed graphs | S. Pirzada et al.

\[M'_{44} = \det \begin{pmatrix} a-1 & -b & -c \\ -a & b-1 & -c \\ -a & -b & c-1 \end{pmatrix} = -4abc + a + b + c - 1.\]

Clearly, the sum of all 3×3 principal minors is M³ = 4a 2 b+4a 2 c+4b ²a+8abc+4c ²a+4b 2 c+ 4c 2 b−4abn−4acn−4bcn−4+3n. Lastly, the determinant of Q is given by det(Q) = 1−n−4a 2 b− 4a 2 c−4b ²a−8abc−4c ²a−4b 2 c−4c 2 b+4abn+4acn+4bcn+16bca²+16acb²+16abc²−16abcn.

Therefore, the characteristic polynomial of Q can be expressed as

\[\Phi_Q(x) = x^4 - \text{Tr}(Q)x^3 + M_2x^2 - M_3x + \det(Q),\]

which after substituting the above values transforms into

\[\Phi_Q(x) = x^4 + (4-n)x^3 + (6-3n)x^2 + C_1x + C_0,\]

where C⁰ and C¹ are same as given in the formulation of statement(ii). It is worth noting that some of the eigenvalues obtained from the quotient matrix may coincide with those already found. To avoid repetition, we must ensure that these eigenvalues are distinct from the previously determined ones.

Next, we prove that −1 is not a root of ΦQ(x) = 0.

\[\Phi_Q(-1) = \det(-I - Q) = (-1)^4 \det(I + Q) = \det(I + Q) = \det\begin{pmatrix} a & -b & -c & n - a - b - c \\ -a & b & -c & n - a - b - c \\ -a & -b & c & n - a - b - c \\ a & b & c & n - a - b - c \end{pmatrix}.\]

Subtracting the 4th row from the first three and expanding along the 4th column, gives

\[\Phi_Q(-1) = -16abc(n - a - b - c) \neq 0.\]

This shows that −1 is not a zero of ΦQ(x) and thus completing the proof of case 1.

Case 2. With a suitable labelling of the vertices of Σ, its adjacency matrix is

\[A(\Sigma) = \begin{pmatrix} J_a - I_a & -J_{a \times b} & -J_{a \times c} \\ -J_{b \times a} & J_b - I_b & -J_{b \times c} \\ -J_{c \times a} & -J_{c \times b} & J_c - I_c \end{pmatrix}.\]

Part (i) is proved by considering a vector w = (x ⊺ , y ⊺ , z ⊺ ) ^⊺ ∈ R n , where x ⊥ ja, y ⊥ j^b and z ⊥ jc. The remaining three eigenvalues are obtained from the 3 × 3 equitable quotient matrix

\[Q = \begin{pmatrix} a - 1 & -b & -c \\ -a & b - 1 & -c \\ -a & -b & c - 1 \end{pmatrix}.\]

Clearly, we have

\[\Phi_Q(x) = x^3 - (n-3)x^2 + (3-2n)x - n + 1 + 4abc\]

and since ΦQ(−1) = 4abc ̸= 0, the proof of the theorem is complete.

As the signed graph under consideration is (n-1)-regular, each eigenvalue \(\lambda\) of \(\Sigma\) yields a Laplacian eigenvalue \(n-1-\lambda\). Hence, the Laplacian spectrum of \(\Sigma\) follows directly from its adjacency spectrum, and the corresponding theorem is omitted. Now, we proceed with the net Laplacian spectrum.

Theorem 3.2. Let \(\Sigma \cong (K_n, K_{a.b,c}^-)\), \(n \neq a+b+c\). The net Laplacian spectrum of \(\Sigma\) is given by

\[\left\{0, (n-2(a+b+c))^2, (n-2b-2c)^{a-1}, (n-2a-2c)^{b-1}, (n-2a-2b)^{c-1}, n^{n-a-b-c}\right\}\]

Proof. With a suitable labelling of the vertices of \(\Sigma\), the diagonal matrix of vertex net degrees is

\[D^{\pm}(\Sigma) = \operatorname{diag}\left((n-2b-2c-1)I_a, (n-2a-2c-1)I_b, (n-2a-2b-1)I_c, (n-1)I_{n-s}\right),\]

where s = a + b + c. Accordingly, the net Laplacian matrix is

\[N(\Sigma) = \begin{pmatrix} (n-2b-2c)I_a - J_a & J_{a\times b} & J_{a\times c} & -J_{a\times (n-s)} \\ J_{b\times a} & (n-2a-2c)I_b - J_b & J_{b\times c} & -J_{b\times (n-s)} \\ J_{c\times a} & J_{c\times b} & (n-2a-2b)I_c - J_c & -J_{c\times (n-s)} \\ -J_{(n-s)\times a} & -J_{(n-s)\times b} & -J_{(n-s)\times c} & nI_{n-s} - J_{n-s} \end{pmatrix}.\]

Let \(\mathbf{x} \in \mathbb{R}^a\) be a vector orthogonal to \(\mathbf{j}_a\). By setting \(\mathbf{z} = (\mathbf{x}^\intercal, \mathbf{o}_b^\intercal, \mathbf{o}_c^\intercal, \mathbf{o}_{n-s}^\intercal)^\intercal \in \mathbb{R}^n\), we immediately obtain \(N(\Sigma)\mathbf{z} = (n-2b-2c)\mathbf{z}\). Therefore, n-2b-2c is the net Laplacian eigenvalue. Since there are a-1 linearly independent column vectors of size a that are orthogonal to \(\mathbf{j}_a\), the algebraic multiplicity of n-2b-2c as an eigenvalue of \(N(\Sigma)\) is (at least) a-1.

By considering a vector \(\mathbf{z} = (\mathbf{o}_a^\intercal, \mathbf{x}^\intercal, \mathbf{o}_c^\intercal, \mathbf{o}_{n-s}^\intercal)^\intercal \in \mathbb{R}^n\), where \(\mathbf{x}\) is one of b-1 vectors that belong to \(\mathbb{R}^b\) and are orthogonal to \(\mathbf{j}_b\), we immediately obtain \(N(\Sigma)\mathbf{z} = (n-2a-2c)\mathbf{z}\).

Similarly, for n-2a-2b, we consider a vector \(\mathbf{z} = (\mathbf{o}_a^\intercal, \mathbf{o}_b^\intercal, \mathbf{x}^\intercal, \mathbf{o}_{n-s}^\intercal)^\intercal \in \mathbb{R}^n\), \(\mathbf{x} \in \mathbb{R}^c\) and \(\mathbf{x} \perp \mathbf{j}_c\), with (c-1) possible linearly independent choices for \(\mathbf{x}\). Finally, \(\mathbf{z} = (\mathbf{o}_a^\intercal, \mathbf{o}_b^\intercal, \mathbf{o}_c^\intercal, \mathbf{x}^\intercal)^\intercal \in \mathbb{R}^n\), where \(\mathbf{x} \in \mathbb{R}^{n-s}\) and \(\mathbf{x} \perp \mathbf{j}_{n-s}\), leads to the analogous conclusion for n.

In order to determine the remaining four net Laplacian eigenvalues, consider a \(4 \times 4\) equitable quotient matrix

\[Q = \begin{pmatrix} n - 2b - 2c - a & b & c & s - n \\ a & n - 2a - 2c - b & c & s - n \\ a & b & n - 2a - 2b - c & s - n \\ -a & -b & -c & s \end{pmatrix}.\]

Observe that \(\mathbf{j}_4\) and \((1,1,1,\frac{-s}{n-s})^{\mathsf{T}}\) are eigenvectors corresponding to 0 and n, respectively. Similarly, by considering two linearly independent vectors \(\mathbf{v}_1 = (-b,a,0,0)\) and \(\mathbf{v}_2 = (-c,0,a,0)\), we immediately obtain \(Q\mathbf{v}_i = (n-2(a+b+c))\mathbf{v}_i, i \in \{1,2\}\). Thus, we conclude that 0 and n-2(a+b+c) are also eigenvalues of \(N(\Sigma)\).

Let \(\lambda_1\) and \(\lambda_2\) be the remaining two net Laplacian eigenvalues of \(\Sigma\). We have

\[\operatorname{Tr}(N(\Sigma)) = a(n-2b-2c-a-1) + b(n-2a-2c-b-1) + c(n-2a-2b-c-1) + (n-a-b-c)(n-1),\]

implies

\[\lambda_1 + \lambda_2 = a(n - 2b - 2c - a - 1) + b(n - 2a - 2c - b - 1) + c(n - 2a - 2b - c - 1) + (n - a - b - c)(n - 1) - (a - 1)(n - 2b - 2c) - (b - 1)(n - 2a - 2c) - (c - 1)(n - 2a - 2b) - (n - a - b - c - 1)n - (n - 2(a + b + c)) = 2(n - a - b - c).\] (1)

Also,

\[\begin{aligned} \operatorname{Tr} \big( N(\Sigma)^2 \big) &= \operatorname{Tr} \Big( ( \big( D^{\pm}(\Sigma) - A(\Sigma) \big) \big( D^{\pm}(\Sigma) - A(\Sigma) \big) \Big) \\ &= \operatorname{Tr} \Big( D^{\pm}(\Sigma)^2 - D^{\pm}(\Sigma) A(\Sigma) - A(\Sigma) D^{\pm}(\Sigma) + A(\Sigma)^2 \Big) \\ &= \operatorname{Tr} ( D^{\pm}(\Sigma)^2 ) - 2 \operatorname{Tr} ( D^{\pm}(\Sigma) A(\Sigma) ) + \operatorname{Tr} ( A(\Sigma)^2 ) \\ &= \sum_{i=1}^n d_i^{\pm^2} + n(n-1), \end{aligned}\]

where the later equality is obtained by using the fact \(\text{Tr}(D^{\pm}(\Sigma)A(\Sigma)) = 0\) and \(\text{Tr}(A(\Sigma)^2) = \sum_{i=1}^{n} d_i\), where \(d_i\) is the degree of the vertex \(v_i\).

Thus, we have

\[\lambda_1^2 + \lambda_2^2 = a(n - 2b - 2c - a - 1)^2 + b(n - 2a - 2c - b - 1)^2 + c(n - 2a - 2b - c - 1)^2 + (n - a - b - c)(n - 1)^2 + n(n - 1) - (a - 1)(n - 2b - 2c)^2 - (b - 1)(n - 2a - 2c)^2 - (c - 1)(n - 2a - 2b)^2 - (n - a - b - c - 1)n^2 - (n - 2(a + b + c))^2 = 2((a + b + c)^2 + (n - (a + b + c))^2).\] (2)

We know that \(2\lambda_1\lambda_2=(\lambda_1+\lambda_2)^2-{\lambda_1}^2-{\lambda_2}^2\) and hence, in view of Equations (1) and (2), we have

\[\lambda_1 \lambda_2 = n \big( n - 2(a+b+c) \big).\]

Thus, \(\lambda_1\) and \(\lambda_2\) satisfy the quadratic equation

\[t^{2} - 2(n - a - b - c)t + n(n - 2(a + b + c)) = 0.\]

Solving this equation gives the roots n and n-2(a+b+c), thereby completing the proof of the theorem.

We illustrate the previous results in an example.

Example 3.3. Consider the complete signed graph \((K_6,K_{2,1,1}^-)\) depicted in Figure 1. By plugging (n,a,b,c)=(6,2,1,1) in Theorem 3.1(case 1), the bi-quadratic equation transforms to \(x^4-2x^3-12x^2+34x-21=0\) and its roots are \(-1\pm2\sqrt{2}\),1,3. Thus, we have

\[\operatorname{Spec}((K_6,K_{2,1,1}^-)) = \Big\{(-1)^2,1,-1-2\sqrt{2},-1+2\sqrt{2},3\Big\},\]

Computation of the eigenvalues of complete signed graphs | S. Pirzada et al.

Figure 1. The signed graph \((K_6, K_{2,1,1}^-)\). Negative edges are dashed.

LSpec(\[(K_6, K_{2,1,1}^-)\]) = \(\{2, 6 - 2\sqrt{2}, 6 + 2\sqrt{2}, 4, 6^2\}\).

Similarly, in view of Theorem 3.2,

NSpec(\[(K_6, K_{2,1,1}^-)\]) = \(\{(-2)^2, 0, 2, 6^2\}\).

4. Adjacency, Laplacian and net Laplacian spectrum of \((K_n, F_{2p+1}^-)\)

Theorem 4.1. Let \(\Sigma \cong (K_n, F_{2p+1}^-)\), \(p \notin \{1, \frac{n-2}{2}, \frac{n-1}{2}\}\). The spectrum of \(\Sigma\) is given by

\[\{(-3)^{p-1}, (-1)^{n-2p-2}, 1^p, \alpha_1, \alpha_2, \alpha_3\},\]where

\(\alpha_i, 1 \leq i \leq 3 \text{ are the roots of } x^3 + (5-n)x^2 + (4p-4n+7)x + 8np - 3n - 16p^2 - 4p + 3 = 0.\)

Proof. With a suitable labelling of the vertices of \(\Sigma\), its adjacency matrix is

\[A(\Sigma) = \begin{pmatrix} 0 & -\mathbf{j}_p^{\mathsf{T}} & -\mathbf{j}_p^{\mathsf{T}} & \mathbf{j}_{n-2p-1}^{\mathsf{T}} \\ -\mathbf{j}_p & J_p - I_p & J_p - 2I_p & J_{p \times (n-2p-1)} \\ -\mathbf{j}_p & J_p - 2I_p & J_p - I_p & J_{p \times (n-2p-1)} \\ \mathbf{j}_{n-2p-1} & J_{(n-2p-1) \times p} & J_{(n-2p-1) \times p} & J_{n-2p-1} - I_{n-2p-1} \end{pmatrix}.\]

Let \(\mathbf{x} \in \mathbb{R}^p\) be a vector orthogonal to \(\mathbf{j}_p\). By setting \(\mathbf{z} = (0, \mathbf{x}^\intercal, \mathbf{x}^\intercal, \mathbf{o}_{n-2p-1}^\intercal)^\intercal \in \mathbb{R}^n\), we immediately obtain \(A(\Sigma)\mathbf{z} = -3\mathbf{z}\). Therefore, -3 is the eigenvalue with multiplicity (at least) p-1.

By setting \(\mathbf{z} = (0, \mathbf{x}^{\mathsf{T}}, -\mathbf{x}^{\mathsf{T}}, \mathbf{o}_{n-2p-1}^{\mathsf{T}})^{\mathsf{T}} \in \mathbb{R}^{n}\), \(x \perp \mathbf{j}_{p}\), we immediately obtain \(A(\Sigma)\mathbf{z} = 1\mathbf{z}\). Therefore, 1 is the eigenvalue with multiplicity (at least) p-1.

Next, we consider a vector \(\mathbf{z} = (0, \mathbf{o}_p^\intercal, \mathbf{o}_p^\intercal, \mathbf{x}^\intercal)^\intercal \in \mathbb{R}^n\), \(x \perp \mathbf{j}_{n-2p-1}\), we immediately obtain \(A(\Sigma)\mathbf{z} = -1\mathbf{z}\). Therefore, -1 is the eigenvalue with multiplicity (at least) n-2p-2.

In order to determine the remaining four eigenvalues, consider a \(4 \times 4\) equitable quotient matrix

\[Q = \begin{pmatrix} 0 & -p & -p & n-2p-1 \\ -1 & p-1 & p-2 & n-2p-1 \\ -1 & p-2 & p-1 & n-2p-1 \\ 1 & p & p & n-2p-2 \end{pmatrix}.\]

The characteristic polynomial of Q is

\[\Phi_Q(x) = (x-1)P(x),\]

where \[P(x) = x^3 + (5-n)x^2 + (4p-4n+7)x + 8np - 3n - 16p^2 - 4p + 3\].

Now, we will show that the roots of P(x) = 0 are distinct from the previously determined ones.

Clearly, \[P(1)=8(2-n+pn-2p^2)=0 \Leftrightarrow p=1 \text{ or } p=\frac{n-2}{2},\] \(P(-1)=0 \Leftrightarrow p=0 \text{ or } p=\frac{n-1}{2} \text{ and }\) \(P(-3)=0 \Leftrightarrow p=0 \text{ or } p=\frac{n-2}{2}.\)

\[P(-1) = 0 \Leftrightarrow p = 0 \text{ or } p = \frac{n-1}{2} \text{ and }\]

\[P(-3) = 0 \iff p = 0 \text{ or } p = \frac{n-2}{2}\]

Since it is given that \(p \notin \{1, \frac{n-2}{2}, \frac{n-1}{2}\}\), it follows that \(P(x) \neq 0\) for \(x \in \{-3, -1, 1\}\). Hence, we conclude that the three roots of P(x) = 0 are also eigenvalues of \(A(\Sigma)\), distinct from those previously determined. The remaining eigenvalue can be determined using the trace of the matrix, since the sum of all eigenvalues equals zero. It is straightforward to verify that this eigenvalue is 1, and therefore its total multiplicity is p. This completes the proof.

As before, we omit the theorem related to the Laplacian spectrum of \(\Sigma\). Finally, we obtain the net Laplacian spectrum of \(\Sigma\).

Theorem 4.2. Let \(\Sigma \cong (K_n, F_{2p+1}^-)\). The net Laplacian spectrum of \(\Sigma\) is given by

\[\{0, (n-6)^p, (n-2)^{p-1}, n^{n-2p-1}, n-4p-2\}.\]

Proof. The diagonal matrix of vertex net degrees is

\[D^{\pm}(\Sigma) = \operatorname{diag}\left((n-4p-1)I_1, (n-5)I_p, (n-5)I_p, (n-1)I_{n-2p-1}\right).\]

Accordingly, the net Laplacian matrix is

\[N(\Sigma) = \begin{pmatrix} n - 4p - 1 & \mathbf{j}_p^\intercal & \mathbf{j}_p^\intercal & -\mathbf{j}_{n-2p-1}^\intercal \\ \mathbf{j}_p & (n-4)I_p - J_p & 2I_p - J_p & -J_{p\times(n-2p-1)} \\ \mathbf{j}_p & 2I_p - J_p & (n-4)I_p - J_p & -J_{p\times(n-2p-1)} \\ -\mathbf{j}_{n-2p-1} & -J_{(n-2p-1)\times p} & -J_{(n-2p-1)\times p} & nI_{n-2p-1} - J_{n-2p-1} \end{pmatrix}.\]

The proof proceeds similarly to that of the preceding theorem i.e., For the net Laplacian eigenvalues n-2, n-6 and n, we select the same vectors as those corresponding to the eigenvalues -3, 1, and -1, respectively, in Theorem 4.1.

Next, we consider \(4 \times 4\) equitable quotient matrix

\[Q = \begin{pmatrix} n - 4p - 1 & p & p & 2p + 1 - n \\ 1 & n - p - 4 & 2 - p & 2p + 1 - n \\ 1 & 2 - p & n - p - 4 & 2p + 1 - n \\ -1 & -p & -p & 2p + 1 \end{pmatrix}.\]

The characteristic polynomial of Q is given by

\[\Phi_Q(x) = x(x-n)(x-n+6)(x-n+4p+2).\]

Therefore, 0 and n-4p-2 are also net Laplacian eigenvalues of \(\Sigma\).

Let \(\lambda_1\) and \(\lambda_2\) be the remaining two net Laplacian eigenvalues of \(\Sigma\). It is not hard to see that

\[\lambda_1 + \lambda_2 = 2n - 6\], \(\lambda_1^2 + \lambda_2^2 = 2(n^2 - 6n + 18)\) and \(\lambda_1 \lambda_2 = n^2 - 6n\).

Computation of the eigenvalues of complete signed graphs | S. Pirzada et al.

Figure 2. The signed graph (K7, F ⁻ 5 ).

Therefore, λ¹ and λ² are the roots of the quadratic equation

\[t^2 - (2n - 6)t + n^2 - 6n = 0,\]

which upon solving yields λ¹ = n and λ² = n − 6, completing the proof.

As before, we provide an example.

Example 4.3. Consider the complete signed graph (K7, F ⁻ 5 ) depicted in Figure 2. By plugging (n, p) = (7, 2) in Theorem 4.1, the cubic equation transforms to x ³ − 2x ² − 13x + 22 = 0 and its roots are -3.5033, 1.6151, 3.8882. Thus, we have

Spec(\[(K_7, F_5^-)\]) = \(\{-3.5033, -3, -1, 1^2, 1.6151, 3.8882\},\\)

LSpec(\[(K_7, F_5^-)\]) = \(\{2.1118, 4.3849, 5^2, 7, 9, 9.5033\}\).

Similarly, in view of Theorem 4.2,

NSpec(\[(K_7, F_5^-)\]) = \(\{-3, 0, 1^2, 5, 7^2\}\).

5. Conclusion

We have examined complete signed graphs Σ in which the set of negative edges induces either a complete tripartite graph or a friendship graph. Note that if −Σ denotes the signed graph obtained by reversing the sign of every edge of Σ , then A(−Σ) = −A(Σ) and N(−Σ) = −N(Σ). Consequently, the corresponding results in the case where the positive edges (instead of the negative ones) induce any of the aforementioned graphs can be derived directly when dealing with the adjacency matrix or the net Laplacian matrix. In particular, the spectrum of −Σ is obtained by negating each eigenvalue of Σ.

On the other hand, L(−Σ) = −L(Σ) does not hold unless Σ is edgeless, since the main diagonal entries remain unchanged. Nevertheless, the analogous results for −Σ can be derived in a similar manner, following the same arguments used for Σ.

Acknowledgements. The authors are grateful to the anonymous referee for the useful comments. The research of S. Pirzada is supported by the NBHM-DAE research project number

NBHM/02011/20/2022. The fourth author is supported by the Indonesia Deposit Insurance Corporation (Lembaga Penjamin Simpanan) under Memorandum of Agreement, Number PKS-3/KPHL/ 2024 and 842/IT1.B07/KS.00/2024.

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