EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 504-518 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spectral Analysis of Splitting Signed Graph Sandeep Kumar1, Deepa Sinha1,∗ 1 Department of Mathematics, South Asian University, New Delhi-110068, India Abstract. An ordered pair Σ = (Σu,σ) is called the signed graph, where Σu = (V,E) is a underly- ing graph and σ is a signed mapping, called signature, from E to the sign set {+,−}. The splitting signed graph Γ(Σ) of a signed graph Σ is defined as, for every vertex u ∈ V (Σ), take a new vertex u′. Join u′ to all the vertices of Σ adjacent to u such that σΓ(u ′v) = σ(u′v), u ∈ N(v). The objective of this paper is to propose an algorithm for the generation of a splitting signed graph, a splitting root signed graph from a given signed graph using Matlab. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and laplacian matrices of the splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the en- ergy of the original signed graph Σ and the energy of the splitting signed graph Γ(Σ) is established. 2020 Mathematics Subject Classifications: 05C22, 05C50, 05C90 Key Words and Phrases: Signed graph, splitting signed graph, spectrum, energy 1. Introduction The initial notation and terminology used in this paper have been sourced from Harary [10], Zaslavsky [20] and West [19]. The graphs examined in this paper are finite and simple. A signed graph, Σ = (Σu, σ), is composed of an underlying graph, Σu = (V,E), where |V | = n & |E| = m, and a signature, σ : E → {+,−}, which labels each edge of Σu as either ‘+’ or ‘−’. In this paper, edges labeled with ‘+’ are considered positive and are depicted using solid lines, while edges labeled with ‘−’ are considered negative and are depicted using dashed lines. If all edges in Σ are signed ‘+’ or ‘−’, the signed graph is referred to as homogeneous, otherwise, it is heterogeneous. Graphs can be thought of as homogeneous signed graphs with each edge being labeled as ‘+’. A cycle in a signed graph Σ is considered positive if it includes an even number of negative edges. If every cycle in Σ is positive, then Σ is defined as balanced signed graph. An ordered pair (Σ, µ) is known as a marked signed graph where Σ = (Σu, σ) is a signed graph, and µ : V (Σu) → {+,−} is a function defined on the vertex set V (Σu) of Σu. The function µ assigns each vertex of Σu to either the positive or negative sign from the set ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4798 Email addresses: ssulakh.94@gmail.com (S. Kumar), deepasinha@sau.ac.in (D. Sinha) https://www.ejpam.com 504 © 2024 EJPAM All rights reserved. S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 505 {+,−}, and is called the marking of Σ. The adjacency matrix of Σ, whose vertices are v1, v2, ..., vn is the n × n matrix A(Σ) = [ai,j ] where ai,j =  0 if vi and vj are not adjacent 1 if σ(vi, vj) is positive −1 if σ(vi, vj) is negative (1) The spectrum of a matrix is a list of its eigenvalues along with their multiplicities. Since, A(Σ) is a symmetric matrix with real entries so all its eigenvalues are real. Let λ1(Σ) > λ2(Σ) > ... > λk(Σ) are distinct eigenvalues of A(Σ) along with their multiplicities m1,m2, ...,mk, 1 ≤ k ≤ n, then the list of eigenvalues of adjacency matrix is called adjacency spectrum of the signed graph Σ and usually denoted as: Sp(Σ) = ( λ1 λ2 · · · λk m1 m2 · · · mk ) Let D(Σ) = [di,j ] be a diagonal matrix of order n such that the entry (i, j) is deg(ui) if i = j and 0 otherwise, where deg(ui) denotes the degree of the vertex ui. D(Σ) is called degree matrix of the signed graph Σ. The Laplacian matrix L(Σ) = [li,j ] of a signed graph Σ is a square matrix of order n such that li,j = di,j − ai,j for 0 ≤ i, j ≤ n. The eigenvalues of the laplacian matrix is called laplacian spectrum and is denoted by SLp(Σ). Two signed graphs are said to be co-spectral if they have same spectrum. The largest eigenvalue λ1(Σ) is called the index of Σ, whereas the largest absolute eigenvalue is called spectral radius ρ(Σ), i.e. ρ = max{λ1(Σ),−λk(Σ)}. (2) The study of graph spectrum is of significant importance in the field of graph theory, and spectral graph-theoretic techniques have been applied in a range of fields including quan- tum physics, chemistry, computer science, and more. In recent years, researchers have explored the spectral properties of graphs constructed through graph operations such as disjoint union, Cartesian product, Kronecker product, strong product, lexicographic prod- uct, corona, edge corona, and neighbourhood corona. A comprehensive overview of results on the spectra of these graphs can be found in the literature [4–7, 9, 14–16]. In [? ] au- thors presented the idea of the splitting graph Γ(Σu) for a given graph Σu. The process of creating the splitting graph Γ(Σu) involves taking a new vertex v′ for each vertex v in graph Σu. The new vertex v′ is then connected to all vertices in Σu that are adjacent to v. The resulting graph is referred to as the splitting graph Γ(Σu) of graph Σu. Recently, a variation of this concept has been applied in the analysis of online social networks (OSNs), where v′ is also connected to v. This variation of the concept is referred to as the “clone” of v. For the purpose of convenience, the term “clone” is adopted for v′ in the splitting graph Γ(Σu) as well. Gutman [11] introduced the concept of energy of a graph Σu in 1978 as the sum of the S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 506 absolute values of its eigenvalues, denoted by E(Σu), i.e., E(Σu) = n∑ i=1 |λi| Later, in 2004, Bapat et.al [3] proved that the energy of a graph can only be an even integer if it is a rational number. Pirzada et.al [12], on the other hand, demonstrated that the energy of a given graph can never be the square root of an odd integer. Graph energy is briefly discussed in [2], while in [18] authors established a relationship between the energy of a graph and its splitting graph. The concept of graph energy has been widely studied in graph theory and has significant applications in various fields. The study of graph energy can provide insights into the structural properties of the graph and is often used in the design and analysis of communication networks, molecular chemistry, and social networks. Additionally, the energy of a graph is closely related to its spectrum and can be used to investigate various graph invariants, such as chromatic number, clique number, and independence number [5, 7, 8, 12, 16]. Sinha et.al [13] introduced the splitting signed graphs as an extension of the splitting graph concept. The splitting signed graph of a signed graph Σ = (V,E, σ), denoted as Γ(Σ) = (VΓ, EΓ, σΓ), is obtained by creating a new vertex v′ for each vertex v ∈ V (Σ), and connecting v′ to all vertices in Σ adjacent to v such that the sign of the corresponding edges is preserved, i.e., σΓ(v ′u) = σ(vu) for all u ∈ N(v). This construction is depicted in Figure 1. A signed graph Σ is called a splitting signed graph if it is isomorphic to the splitting signed graph Γ(U) of some signed graph U , where U is referred to as the splitting root signed graph of Σ. 1 2 3 4 5 1 2 1 ' 2' 3' 4' 5' 35 4 Figure 1: Signed graph Σ and its splitting signed graph Γ(Σ) Algorithmic characterization of splitting signed graph by Sinha et.al appears in the proceedings of the International Conference on Current Trends in Graph Theory and Computation in [17]. Here, in this research paper, we give an algorithm to generate the splitting signed graph and its variant, the splitting root signed graph, from a given signed graph by using Matlab. We also perform spectral analysis on the adjacency and Lapla- cian matrices of the splitting signed graph to investigate its eigenvalues and eigenvectors. Furthermore, we establish a relationship between the energy of the original signed graph and the energy of the splitting signed graph. Our study sheds light on the properties of S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 507 the splitting signed graph, and its potential applications in various domains. The Kronecker product (or tensor product) of matrices U and W is a matrix defined as follows: U ⊗W =  a11W a12W · · · a1,nW a21W a22W · · · a2,nW ... ... . . . ... am1W am2W · · · am,nW  where, U ∈ Rm×n and W ∈ Rp×q. Theorem 1. [8] Let U and W are two square matrices such that U ∈ Mm and W ∈ Mn. If µi is an eigenvalue of U with its corresponding eigenvector yi, and λj is an eigenvalue of W with its corresponding eigenvector xi, then µiλj is an eigenvalue of the Kronecker product U ⊗W , with the corresponding eigenvector yi ⊗ xj. 2. Generating splitting signed graph The procedure for generating a splitting signed graph can be described as follows: Given a graph with n vertices, the first step is to encode an n × n symmetric adjacency matrix for the graph. Since a new vertex is created for each vertex in the original graph, the splitting graph will have a total of 2n vertices. The non-zero entries in the first row of the adjacency matrix indicate the vertices which are adjacent to the first vertex i.e. v1. These entries in first row are also considered adjacent to vertex vn+1 and are updated in the output matrix. This process is repeated for each row until all rows have been processed. As a result, a 2n × 2n output matrix is generated. The adjacency matrices of the original signed graph Σ and its splitting signed graph Γ(Σ) can be represented as follows: A(Σ) =  0 1 0 0 0 1 0 1 0 −1 0 1 0 1 0 0 0 1 0 −1 0 −1 0 −1 0  and A(Γ(Σ)) =  0 1 0 0 0 0 1 0 0 0 1 0 1 0 −1 1 0 1 0 −1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 −1 0 0 1 0 −1 0 −1 0 −1 0 0 −1 0 −1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 −1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 −1 0 −1 0 0 0 0 0 0  S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 508 It has been observed that the adjacency matrix, of order 2n, of the splitting signed graph Γ(Σ) can be partitioned into four equal matrices each of order n, where the initial three matrices are identical and the fourth is always zero. Additionally, it has been discovered that the original signed graph Σ is an induced subgraph of its splitting signed graph Γ(Σ). Algorithm 1 Algorithm to derive the splitting signed graph Γ(Σ) of a signed graph Σ 1: Input: number of vertices (N) 2: Input: adjacency matrix of the signed graph A(Σ) = a(i, j) 3: for i = 1 : 2N do 4: for j = 1 : 2N do 5: check 6: if i > N&&j > N then 7: Assign a(i,j)=0; 8: else 9: check 10: if i > N&&j ≤ N then 11: Assign a(i,j) = a(i-N,j); 12: else 13: check 14: if i ≤ N&&j > N then 15: Assign a(i,j) = a(i,j-N); 16: else 17: Assign a(i,j) = a(i,j); 18: end if 19: end if 20: end if 21: end for 22: end for 23: Output Generate 2N × 2N matrix i.e. adjacency matrix of signed split graph Computational complexity: Computational complexity analysis is an essential aspect of evaluating the performance of an algorithm. In this regard, we analyze the complexity involved in Steps 3 and 4 of our algorithm. In these steps, we traverse each vertex of the signed graph and examine its adjacency with all other vertices. As a result, the complexity involved in these steps is O(n2). Considering the overall algorithm, the total complexity involved is the sum of the complexities of all the steps. As the complexity in Steps 3 and 4 is the highest, the total complexity is also O(n2). Therefore, the complexity of the proposed algorithm for finding a signed split graph with a given adjacency matrix is O(n2), where n represents the number of vertices in the signed graph. S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 509 3. Structural characterization to derive splitting root signed graph Sampathkumar and Walikar [? ] presented the following characterization of splitting graphs, which is utilized to derive the splitting root graph. Theorem 2. [? ] A graph Σu can be characterized as a splitting graph if and only if its vertex set, V (Σu), can be divided into two sets, V1 and V2, such that: (a) there exists a bijective mapping between V1 and V2, with v mapping to v′, and (b) the neighbours of v′, N(v′), are equal to the intersection of the neighbours of v, N(v), and the set V1. In [17] authors presented a structural characterization of a splitting signed graph that can be used to derive the splitting root signed graph. Theorem 3. [17] Let Σ be a connected signed graph, then Σ is splitting signed graph if and only if the following two conditions hold: (a) The underlying graph Σu is splitting graph (b) the vertex set of Σ can be divided into two sets, V1 and V2, such that for each u′ ∈ V2 there exist a vertex u ∈ V1 that holds the condition σ(u′v) = σ(uv) for every v ∈ V1 ∩N(v). Generating splitting root signed graph using Theorem 3 The process for deriving the splitting root signed graph involves several steps. Firstly, it is necessary for the number of vertices, denoted as “n”, to be even in order for a splitting root signed graph to exist. If “n” is odd, then it is impossible to construct a splitting root signed graph. Once it has been established that “n” is even, an n dimensional matrix is generated to represent the given signed graph. The matrix is examined to count the number of positive and negative edges, and it is determined whether both counts are divisible by 3. If this condition is met, a splitting root signed graph can be created; otherwise, it cannot. The next step involves calculating the number of negative and positive edges for each vertex by counting the number of −1s and 1s in each row. If it is possible to partition the vertex set V (Σ) into two sets, such that the number of negative and positive edges in one set is exactly double of the other set, then a splitting root signed graph can be constructed. If a splitting root signed graph exists, it’s adjacency matrix of order n can be partitioned into four matrices of equal order n 2 . Let [ai,j ], [bi,j ], [ci,j ] and [di,j ] are the four matrices of order n 2 , then ai,j = bi,j = ci,j for 0 ≤ i, j ≤ n 2 and di,j = 0 for 0 ≤ i, j ≤ n 2 i.e. [di,j ] is zero matrix. To make the matrix identical and zero, it is necessary to interchange rows and columns. This will result in a matrix of size n 2 × n 2 . An example of the computation of a splitting root signed graph from a given signed graph will be provided in the following section. S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 510 A(Σ1) =  0 −1 0 −1 −1 0 −1 0 −1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 −1 0 1 0 0 0 0 0 −1 0 1 0 0 1 0 −1 0 0 0 0 1 0 1 0 −1 0 1 0 0 1 0 −1 0 0 0 0 −1 0 −1 0  The adjacency matrix of this signed graph, A(Σ1) is a 8× 8 matrix. Here the number of vertices in a signed graph is even, so we can find its splitting root signed graph. However, if the vertices are odd in numbers, then splitting root signed graph of the given signed graph will not exist. In this case, there are 12 occurrences of the value −1 and 12 occurrences of the value 1 in the adjacency matrix. Clearly, here both the values are divisible by 3, so the splitting root signed graph can be computed. The next step is to count the number of 1s and −1s in each row of the matrix: Vertex vi no. of 1s no. of −1s 1 0 4 2 1 1 3 4 0 4 1 1 5 2 2 6 2 0 7 2 2 8 0 2 With this knowledge, the vertex set is splitted into two sets, V1 and V2, where the number of 1s and −1s in V1 is double that of in V2. In this example, V1 is {1, 3, 5, 7} and V2 is {2, 4, 6, 8}. If the partition is successful, the splitting root signed graph exists, otherwise it does not. Next, the splitting root graph is computed. The adjacency matrix A(Γ(Σ)), of order 2n, of the splitting signed graph can be partitioned into four equal matrices of the order n. Let [ai,j ], [bi,j ], [ci,j ] and [di,j ] are the four matrices of order n, then ai,j = bi,j = ci,j for 0 ≤ i, j ≤ n and di,j = 0 for 0 ≤ i, j ≤ n i.e. [di,j ] is zero matrix. In this example, the 8 × 8 input matrix is divided into four equal matrices of size 4× 4. The fourth matrix is made zero and the first, second, and third matrices are made identical through row and column transformations. The output matrix is 4× 4. After applying transformations such that R2 ⇔ R7 and C2 ⇔ C7, we get: S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 511 Vertex vi 1 7 3 4 5 6 2 8 1 0 -1 0 -1 -1 0 -1 0 7 -1 0 1 0 0 1 0 -1 3 0 1 0 1 1 0 1 0 4 -1 0 1 0 0 0 0 0 5 -1 0 1 0 0 1 0 -1 6 0 1 0 0 1 0 0 0 2 -1 0 1 0 0 0 0 0 8 0 -1 0 0 -1 0 0 0 and then R4 ⇔ R5 and C4 ⇔ C5, we get: Vertex vi 1 7 3 4 5 6 2 8 1 0 -1 0 -1 -1 0 -1 0 7 -1 0 1 0 0 1 0 -1 3 0 1 0 1 1 0 1 0 4 -1 0 1 0 0 1 0 -1 5 -1 0 1 0 0 0 0 0 6 0 1 0 1 0 0 0 0 2 -1 0 1 0 0 0 0 0 8 0 -1 0 -1 0 0 0 0 The first, second, and third matrices can be seen to be the same, and the fourth matrix is zero, which satisfies the requirements for a splitting root signed graph. The final output matrix is the splitting root signed graph: Vertex vi 1 7 3 5 1 0 -1 0 -1 7 -1 0 1 0 3 0 1 0 1 5 -1 0 1 0 The splitting signed graph S1 and its corresponding splitting root signed graph S are depicted in the Figure 2. 4 2 S1 7 8 6 5 3 1 S Figure 2: Signed graph S1 and S is its splitting root signed graph S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 512 Algorithm 2 Algorithm to derive the splitting root signed graph of a given signed graph 1: Input: number of vertices (N) 2: Input: adjacency matrix of the signed graph A(Σ) = a(i, j) 3: Check 4: if N mod 2 = 1 then 5: Print: matrix is of odd order, splitting root signed graph dose not exist 6: Terminate 7: else 8: for i = 1 : N do 9: for j = 1 : N do 10: check 11: if a(i,j)==1 then 12: countp = countp + 1 13: else 14: if a(i,j)== -1 then 15: countq = countq + 1 16: end if 17: end if 18: end for 19: end for 20: end if 21: check 22: if countp mod 3 == 0 && countq mod 3 == 0 then 23: Print: Splitting root signed graph is possible 24: else 25: Print: Splitting root signed graph is not possible and Terminate 26: end if 27: Assign: positive[i] = countp 28: negative[i] = countq 29: boole[i] = F 30: for i = 1 : N do 31: for j = i+ 1 : N do 32: Assign: k = i 33: if boole[i] == T then 34: no-found = 1 35: if positive[i] == 2 positive[j] && negative[i] == 2 negative[j] && boole[i] == T then S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 513 36: Assign: vertex[i] = 2 37: vertex[j] = 1 38: boole[i] = T 39: boole[i] = T 40: no-found = 0 41: end if 42: if positive[i] ==positive[j] 2 && negative[i] ==negative[j] 2 && boole[i] == T then 43: Assign: vertex[i] = 1 44: vertex[j] = 2 45: boole[i] = T 46: boole[i] = T 47: no-found = 0 48: end if 49: if no-found == 0 then 50: Assign: j=n 51: if no-found == 0 then 52: Print: Splitting graph is not possible as vertex division is not proper and Terminate 53: end if 54: end if 55: for i = 1 : N do 56: if vertex[i] == 2 then 57: Assign: arrayhigh[o1] = i 58: count o1 = o1+1 59: else 60: if vertex[i] == 1 then 61: Assign: arrayhigh[o2] = i 62: count o2 = o2+1 63: end if 64: end if 65: end for 66: end if 67: Assign: j = 1 68: for i = 1 : N 2 do 69: Assign: val = arrayhigh[i] 70: if val > N 2 then 71: Assign: val-low = arraylow[j] 72: if val − low ≤ n 2 then 73: Apply row transformation 74: end if 75: end if 76: end for 77: Assign: j = j+1 78: for i = 1 : 2N do 79: for j = 1 : 2N do S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 514 80: Assign: b(i,j) = a(i,j) 81: Divide the matrix into four equal blocks 82: Set: val-ret = Above divided matrix 83: if val-ret == 0 then 84: Assign: j=N+1 and Terminate 85: else 86: Assign: a(i,j) = b(i,j) 87: end if 88: end for 89: end for 90: end for 91: end for 92: Output: Generate [ai,j ] matrix of order N 2 Computational complexity: The algorithm calculates the total number of positive and negative edges and traverses every entry of the matrix. As a result, the complexity to count the edges is O(n2). In Steps 30 to 52, the vertex set is partitioned into two sets such that number of positive and negative edges in one set is exactly double of the other. As a result, the complexity is O(n2). If the function is denoted by fun then the complexity required for the row and column transformations from Step 67 to Step 77 to make all the three matrices identical is O(n3). If the fourth submatrix is already zero in the steps 78 to 91 then we apply row operation to make all other sub matrices identical. We traverse each vertex of the signed graph and examine every entry if it is identical. As a result, the complexity involved in these steps is O(n2×n×n) = O(n4). Therefore, the complexity of the proposed algorithm for finding a root signed split graph with a given adjacency matrix is O(n2)+O(n3)+O(n4)+O(n2) = O(n4)., where n represents the number of vertices in the signed graph. 4. Spectrum of splitting signed graph In this section, we aim to find out the spectrum of the Adjacency matrix and Laplacian matrix of a splitting signed graph Γ(Σ). Let A(Σ) be the adjacency matrix of the signed graph Σ on n vertices, and is given as follow A(Σ) =  0 a1,2 · · · a1,n a2,1 0 · · · a2,n ... ... . . . ... an,1 an,2 · · · 0  Let v′i be the vertex corresponding to vi, 1 ≤ i ≤ n, which is added in Σ to construct Γ(Σ), such that N(v′i) = N(vi), for 1 ≤ i ≤ n. Then the adjacency matrix of Γ(Σ), A(Γ(Σ)), S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 515 can be expressed as a block matrix with blocks as follows A(Γ(Σ)) =  0 a1,2 · · · a1,n 0 a1,2 · · · a1,n a2,1 0 · · · a2,n a2,1 0 · · · a2,n ... ... . . . ... ... ... . . . ... an,1 an,2 · · · 0 an,1 an,2 · · · 0 0 a1,2 · · · a1,n 0 0 · · · 0 a2,1 0 · · · a2,n 0 0 · · · 0 ... ... . . . ... ... ... . . . ... an,1 an,2 · · · 0 0 0 · · · 0  Let λ1(Σ), λ2(Σ), ..., λn(Σ) are the eigenvalues of the signed graph Σ. We can write above adjacency matrix as, A(Γ(Σ)) = [ A(Σ) A(Σ) A(Σ) 0 ] = [ 1 1 1 0 ] ⊗A(Σ) From here we can see that adjacency matrix A(λ(Σ)) is a Kronecker product of the ma- trices M and A(Σ), where M = [ 1 1 1 0 ] . Easily we can see that {1+ √ 5 2 , 1− √ 5 2 } are the eigenvalues of the matrix M . So the adjacency spectrum of the splitting signed graph λ(Σ) is given as {(1+ √ 5 2 )λi, ( 1− √ 5 2 )λi}, where 1 ≤ i ≤ n. Now consider L(Σ) be the laplacian matrix of the signed graph Σ and ψ1, ψ2, ..., ψi are the eigenvalues of the laplacian matrix L(Σ). By some easy calculations we can find that the laplacian matrix L(Γ(Σ)) of the splitting signed graph is a Kronecker product of the matrices M and L(Σ), where M = [ 1 1 1 0 ] . So the laplacian spectrum of the splitting signed graph λ(Σ) is given as {(1+ √ 5 2 )ψi, ( 1− √ 5 2 )ψi}, where 1 ≤ i ≤ n. From Equation 2, we can see that spectral radius of Σ will always be greater than equal to the index , i.e. ρ(Σ) ≥ λ1(Σ) In [1] Acharya provided the spectral criterion for balance in Σ as, Theorem 4. [1] A signed graph Σ is balanced if and only if it is cospectral to it’s underlying graph. It provides that the balanced signed graphs have the spectral radius equal to the index. So, here in this section we characterize the balanced splitting signed graphs. Sampathkumar provided an important characterization of balanced signed graphs based on marking: S. Kumar, D. Sinha / Eur. J. Pure Appl. Math, 17 (1) (2024), 504-518 516 Theorem 5. [? ] The balance of a signed graph Σ = (Σu, σ) can be determined if and only if there is a marking µ of its vertices such that the sign of each edge vu in Σ satisfies the condition σ(vu) = µ(v)µ(u). The operation of changing the sign of every edge in a signed graph Σ to its opposite, based on the marking µ of its vertices, is called switching Σ with respect to µ. This operation is performed whenever the end vertices of an edge have opposite signs in Σµ. The concept of switching signed graphs is closely connected to the concept of balance, as indicated by the following theorem: Theorem 6. [20] A signed graph Σ = (Σu, σ) is considered balanced if and only if it is equivalent under switching to its underlying graph Σu. Theorem 7. [13] The splitting signed graph Γ(Σ) is balanced if and only if the signed graph Σ is balanced. Remark 1. The splitting signed graph Γ(Σ) is cospectral to it’s underlying graph Γ(Σu) if and only if Σ is balanced. 5. Energy of splitting signed graph In this section, we explore the connection between the energy of a signed graph Σ and its splitting signed graph Γ(Σ). Theorem 8. Let E(Σ) be the energy of signed graph Σ and E(Γ(Σ)) be the energy of splitting signed graph Γ(Σ), then E(Γ(Σ)) = √ 5E(Σ). Proof. Let V = {v1, v2, ..., vn} be the vertex set of the signed graph Σ. Then adjacency matrix of Σ is given by, A(Σ) =  0 a1,2 · · · a1,n a2,1 0 · · · a2,n ... ... . . . ... am,1 am,2 · · · 0  Let v′i be the vertex corresponding to vi, 1 ≤ i ≤ n, which is added in Σ to construct Γ(Σ), such that N(v′i) = N(vi), for 1 ≤ i ≤ n. Then the adjacency matrix of Γ(Σ), A(Γ(Σ)), can be expressed as a block matrix with blocks as follows A(Γ(Σ)) =  0 a1,2 · · · a1,n 0 a1,2 · · · a1,n a2,1 0 · · · a2,n a2,1 0 · · · a2,n ... ... . . . ... ... ... . . . ... an,1 an,2 · · · 0 an,1 an,2 · · · 0 0 a1,2 · · · a1,n 0 0 · · · 0 a2,1 0 · · · a2,n 0 0 · · · 0 ... ... . . . ... ... ... . . . ... an,1 an,2 · · · 0 0 0 · · · 0  REFERENCES 517 We can write it as, A(Γ(Σ)) = [ A(Σ) A(Σ) A(Σ) 0 ] = [ 1 1 1 0 ] ⊗A(Σ) Let λ1, λ2, ..., λn are the eigenvalues of the signed graph Σ and we can observe that the eigenvalues of [ 1 1 1 0 ] are {1+ √ 5 2 , 1− √ 5 2 }. Therefore, spec(Γ(Σ)) = ( (1+ √ 5 2 )λi (1− √ 5 2 )λi n n ) E(Γ(Σ)) = n∑ i=1 |(1± √ 5 2 )λi| = n∑ i=1 |λi|[ 1 + √ 5 2 + 1− √ 5 2 ] = √ 5 n∑ i=1 |λi| Hence, E(Γ(Σ)) = √ 5E(Σ) 6. Conclusion and Scope In conclusion, this research gives an algorithm for generating a splitting signed graph and a splitting root signed graph from a given signed graph, provided it exists. Addition- ally, a spectral analysis of the resulting graph is conducted by studying its eigenvalues and eigenvectors through the adjacency and laplacian matrices. The research also establishes a relationship between the energy of the original signed graph Σ and the energy of the splitting signed graph Γ(Σ). The scope of this research could be extended by exploring further applications of the proposed algorithm and studying the properties of splitting signed graphs in more detail. Conflicts of Interest: All the authors declare that they have no conflicts of interest regarding the publication of this paper. References [1] B Devadas Acharya. Spectral criterion for cycle balance in networks. Journal of Graph Theory, 4(1):1–11, 1980. [2] R Balakrishnan. The energy of a graph. 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