This study delves into the nuanced dynamics of the Iterated Local Anti-Transitivity (ILAT) model in the realm of signed complex networks. A signed network is derived from a standard network by assigning a sign, either positive or negative, to each edge. The ILAT model for signed networks is based on the concept of an anti-clone vertex, which serves as the counterpart to the clone vertex in the Iterated Local Transitivity (ILT) model. The framework works on the notion of the anti-neighbourhood property of a vertex v. It is denoted by N(v)¯, and is given as N(v)¯={w:w∈V(Σ)∖N[v]}, where V(Σ) is the vertex set of Σ, N(v) is the open neighbourhood of vertex v, and N[v] denotes closed neighbourhood of vertex v. For each vertex v in the given signed network, a corresponding anti-clone vertex v∗ is introduced, which is adjacent to every vertex in N(v)¯. Each newly created edge v∗w, where w∈N(v)¯, is assigned a sign according to the relative number of positive and negative anti-neighbours of v. Furthermore, this study examines the structural and relational properties of the anti-clone vertices as they are incrementally added to the network at each discrete time step t, thereby shedding light on the complex dynamics that evolves through this iterative process.

A signed graph Σ is a pair Σ=(Σu,σ)that consists of a graph (Σu,E) and a sign mapping called signature σ from E to the sign group {+,−}. In this paper, we discuss the t-path product signed graph (Σ)^twhere vertex set of (Σ)^t is the same as that of Σ and two vertices are adjacent if there is a path of length t, between them in the signed graph Σ. The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph Σ, where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of Σ which are switching equivalent to t-path product signed graphs (Σ)^t for t=2,3which are switching equivalent to Σand also the negation of the signed graph ŋ(Σ) that are switching equivalent to (Σ)^t for t=2,3. We also characterize signed graphs that are switching equivalent to t-distance signed graph (Σ¯)t for t=2 where 2-distance signed graph (Σ¯)2=(V′,E′,σ′) defined as follows: the vertex set is same as the original signed graph Σ and two vertices u,v∈(Σ¯)2, are adjacent if and only if there exists a distance of length two in Σ. The edge uv∈(Σ¯)2 is negative if and only if all the edges, in all the distances of length two in Σ are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks. • t-path product signed graph. • t-distance signed graph

The encryption and decryption of sensitive information have become increasingly important in today's digital age. In this paper, we explore the innovative application of signed Cayley graphs for achieving enhanced security in data transmission. In the realm of communication, where information exchange is pervasive, the graph itself serves as the message, with its vertices, edges, and edge signatures containing the concealed information. This paper presents the development of an encryption and decryption scheme that utilizes a signed Cayley graph, along with a private (symmetric) key, to securely encrypt and decrypt the edges of the graph.

A subset C of the vertex set of a graph Gamma is called a perfect code of Gamma if every vertex of Gamma is at a distance of no more than one vertex in C. The biclique partition number of a graph Gamma is the minimum number of complete bipartite subgraphs (bicliques) required to partition the edge set of Gamma. The decision form of the minimum biclique cover problem is classified as NP-Complete. Let G be a finite abelian group and S be any subset of G. The Cayley sum graph Gamma(G,S) is a simple graph with G as its vertex set, and two vertices u and v are adjacent if u + v is an element of S. In this paper, we give some conditions on subset S of G for finding the perfect code set and inequality of biclique partition number of Cayley sum graph Gamma(G,S) and Cayley sum signed graph Gamma(Sigma).

An ordered pair sigma = (sigma 𝑢�, sigma) is called the signed graph, where sigma 𝑢� = (𝑉� , 𝐸�) is an underlying graph and 𝜎� is a signed mapping, called signature, from 𝐸� to the sign set {+, -}. A marking of sigma is a function 𝜇� : 𝑉� (sigma) -> {+, -}.The canonical marking of a signed graph sigma, denoted 𝜇�𝜎� , is given as 𝜇�𝜎� (𝑣�) = pi 𝑣�𝑢�is an element of 𝐸�(sigma)𝜎�(𝑣�𝑢�). The canonical splitting signed graph 𝜉�(sigma) of a signed graph sigma is defined as a signed graph 𝜉�(sigma) = (𝑉� (𝜉�), 𝐸�(𝜉�)) , with 𝑉� (𝜉�) = 𝑉� (sigma) boolean OR 𝑉� ' , where 𝑉� ' is copy of a vertex set in 𝑉� (sigma) s.t. for each vertex 𝑢� is an element of 𝑉� (sigma), take a new vertex 𝑢�' and 𝐸�(𝜉�) is defined as, join 𝑢�' to all the vertices of sigma adjacent to 𝑢� by negative edge if 𝜇�𝜎� (𝑢�) = 𝜇�𝜎� (𝑣�) = -, where 𝑣� is an element of 𝑁�(𝑢�) and by positive edge otherwise. The objective of this paper is to propose an algorithm for the generation of a canonical splitting signed graph, a splitting root signed graph from a given signed graph, provided it exists and to give the characterization of balanced canonical splitting signed graph. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and Laplacian matrices of the canonical splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph sigma and the energy of the canonical splitting signed graph 𝜉�(sigma) is established. Algorithm to generate canonical splitting signed graph 𝜉�(sigma).Spectral Analysisis performed for both adjacency and Laplacian matrices of canonicalsplitting signed graph 𝜉�(sigma).

The Cayley sum graph is a graph whose vertex comprises elements of an abelian group G and edges are the sum of these vertices belonging to a subset of G, namely, S. We introduce Cayley sum signed graph by giving the sign to these edges. An edge receives a positive sign if any of the incident vertices belong to S; otherwise, it receives a negative sign. We discuss the balance, clusterability and some properties of derived Cayley sum signed graph.

Let [Formula: see text] be the ring of integer modulo [Formula: see text] with two binary operators, addition [Formula: see text] and multiplication [Formula: see text], where [Formula: see text] is a positive integer. The special set [Formula: see text] is defined as [Formula: see text]. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set [Formula: see text] and denote it by [Formula: see text]. In this paper, we define a relationship between [Formula: see text] and [Formula: see text], [Formula: see text] is a derived graph from [Formula: see text] by removing [Formula: see text] edges, where [Formula: see text] is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and [Formula: see text]. We also provide values of [Formula: see text] for which the graph [Formula: see text] is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness.

A signed graph Σ is an ordered pair (G, σ ) that consists of a underlying graph G=(V,E) and a sign mapping called signature σ from E to the sign set { +, - } . In this article, we provide another way of looking at the Cartesian product of a path graph and an arbitrary signed graph Σ . We then present the adjacency spectrum and Laplacian spectrum of the Cartesian product in terms of the spectrum and Laplacian spectrum of Σ , respectively. We further provide an upper bound and lower bound for the respective energies. As applications, the results in this article are used (1) to construct a family of infinitely many cospectral and Laplacian cospectral graphs and (2) to compute the adjacency (respectively, Laplacian) spectrum of some known classes of graphs.

An ordered pair $\Sigma = (\Sigma^{u}$,$\sigma$) is called the \textit{signed graph}, where $\Sigma^{u} = (V,E)$ is a \textit{underlying graph} and $\sigma$ is a signed mapping, called \textit{signature}, from $E$ to the sign set $\lbrace +, - \rbrace$. The \textit{splitting signed graph} $\Gamma(\Sigma)$ of a signed graph $\Sigma$ is defined as, for every vertex $u \in V(\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\Sigma$ adjacent to $u$ such that $\sigma_{\Gamma}(u'v) = \sigma(u'v), \ u \in N(v)$. 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 energy of the original signed graph $\Sigma$ and the energy of the splitting signed graph $\Gamma(\Sigma)$ is established.

In complex real-world networks, the relation among vertices (people) changes over time. Even with millions of vertices, adding new vertices or deleting a few previous ones can drastically change the network’s dynamics. The Iterated Local Transitivity model is a deterministic model based on the principle of transitivity and local interaction among people. The same has been extended to signed social networks. Let Σ be a signed graph with underlying graph G = (V, E) and a function σ :E→{+,-} assigning signs to the edges. We determine the relation between the characteristic polynomials of signed graph Σ and the signed graph obtained from Σ by adding (deleting) vertices or by adding (deleting) edges. Consequently, we present a recurrence relation for a characteristic polynomial of the Iterated Local Transitivity model for signed graphs.

The concept of associating a graph to a ring was initiated by Beck in 1988. The zero-divisor graph of a commutative ring [Formula: see text] with unity ([Formula: see text]) is defined as a simple graph, denoted by [Formula: see text], where elements of the ring [Formula: see text] represent vertices and two distinct vertices in [Formula: see text] are adjacent if the product of the corresponding elements is zero in [Formula: see text]. Here, we extend the notion of zero-divisor graphs to signed graphs. We introduce four types of signing to the zero-divisor graphs. Further, we characterize rings for which these four types of signed graphs, its lined signed graphs and their negations are balanced, clusterable, sign-compatible, canonically sign-compatible, and canonically consistent.

Information security and confidentiality are burning issues in today's digitally connected world. Encryption is considered a fast-moving trend in this reference. Although we see rigorous enhancements in the development of information security tools, storage and retrieval, enormous challenges still occur due to unsecured transmission channels, hacking tools and the ubiquity of the Internet. This paper proposes a new cryptography mechanism for secure data transmission and retrieval using a signed graph, its adjacency matrix and the RSA algorithm.

We define an addition signed Cayley graph on a unitary addition Cayley graph Gn represented by Σn∧, and study several properties such as balancing, clusterability and sign compatibility of the addition signed Cayley graph Σn∧. We also study the characterization of canonical consistency of Σn∧, for some n.

We define the signed Cayley graph on Cayley graph X-n denoted by S-n, and study several properties such as balancing, clusterability and sign-compatibility of the signed Cayley graph S-n. Apart from it we also study the characterization of the canonical consistency of S-n, for some n.

For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo n, denoted by AG(Zn). We carried out computer experiments and provide a program for the same. We further characterize cases in which AG(Zn), its complement AG(Zn)¯, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring R with unity.

Common-Edge signed graph CE(S) of a signed graph S is a signed graph whose vertex-set is the pairs of adjacent edges in S and two vertices are adjacent if the corresponding pairs of adjacent edges of S have exactly one edge in common, with the sign same as that of Common-Edge. S -Marked signed graph T is a signed graph which receives the marking μ due to the signed graph S called marker. Further, T is S -consistent if a marker S is defined and if S -markingμ of T with respect to which marked signed graph Tμ is consistent. In this paper, we give an algorithm to detect if CE(S) is S -consistent or not and determine its complexity. • Algorithm to detect if CE(S) is S -consistent or not. • Determination of algorithm's complexity.

ABSTRACT A is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph S is called sign-regular if is same for all and is same for all . The problems of embedding -sign-regular signed graphs in -sign-regular signed graphs is one of the fascinating problem from application point of view which is dealt in this paper with insertion of least number of vertices in S and the problem of finding least number of non-isomorphic co-regular signed graphs is explored. We also define the relationship between characteristic polynomial of graph G and the graph in which it embeds.

For a commutative ring R with unity ( $$1\ne 0$$ ), the zero-divisor graph of R is a simple graph with vertices as elements of $$Z(R)^{*}=Z(R)\setminus \{ 0 \}$$ , where Z(R) is the set of zero-divisors of R and two distinct vertices are adjacent whenever their product is zero. An algorithm is presented to create a zero-divisor graph for the ring of Gaussian integers modulo $$2^{n}$$ for $$n\ge 1$$ . The zero-divisor graph $$\Gamma (\mathbb {Z}_{2^{n}}[i])$$ can be expressed as a generalized join graph $$G[G_{1}, \dots , G_{j}]$$ , where $$G_{i}$$ is either a complete graph (including loops) or its complement and G is the compressed zero-divisor graph of $$\mathbb {Z}_{2^{2n}}$$ . Next, we show that the number of isomorphisms between the zero-divisor graphs for the ring of Gaussian integers modulo $$2^{n}$$ and the ring of integers modulo $$2^{2n}$$ is equal to $$\prod _{j=1}^{2(n-1)}2^{j}!$$ .

In this article, we focus on the characteristic polynomial of a graph containingloops, but without multiple edges. We present a relationship between thecharacteristic polynomial of a graph with loops and the graph obtained byremoving all the loops. In turn, we compute the characteristic polynomial ofunitary addition Cayley graphs.