In this paper, we have characterized the commutative rings with unity for which line Signed Graph of a Signed unit Graph is balanced and consistent. To do this, we first establish some sufficient conditions for balance and consistency of line Signed Graph of Signed unit Graphs.

The order sum Graph associated with the group $G$, denoted by $\Gamma_{os}$, is a Graph with vertex set consisting of elements of $G$ and two vertices say $a$,$b$ $\in \Gamma_{os}$ are adjacent if $o(a)+o(b)>o(G)$, where $o(\ast)$ denotes the order of a group or an element of a group. In this paper, we introduce a Signed Graph called order sum Signed Graph where the underlying Graph is a complete Graph of order $n$ and the edges receive positive and negative signs based on the order sum Graph. We characterise the balanced negated order sum Signed Graphs. We also characterise the positive and negative homogeneous order sum Signed Graphs. Further, we study the properties such as clusterability, sign-compatibility, consistency and switching of Signed Graphs. Further, we obtain the adjacency spectra, Laplacian spectra and signless Laplacian spectra of the order sum Signed Graphs associated with cyclic groups.

Let G be a Signed Graph, where G = (V; E) is the underlying simple Graph and  : E(G) → {± 1} is the sign function on E(G). In this paper, we obtain k-th Signed spectral moments and k-th Signed Laplacian spectral moments of a Signed Graph G , together with coe cients of their Signed characteristic polynomial and Signed Laplacian characteristic polynomial are calculated.

Let k  1 be an integer. A weak Signed Roman k-dominating function on a Graph G is a function f: V (G)! f1; 1; 2g such that P u2N[v] f(u)  k for every v 2 V (G), where N[v] is the closed neighborhood of v. A set ff1; f2; : : :; fdg of distinct weak Signed Roman k-dominating functions on G with the property that Pd i=1 fi(v)  k for each v 2 V (G), is called a weak Signed Roman k-dominating family (of functions) on G. The maximum number of functions in a weak Signed Roman kdominating family on G is the weak Signed Roman k-domatic number of G, denoted by d k wsR (G). In this paper we initiate the study of the weak Signed Roman k-domatic number in Graphs, and we present sharp bounds for d (G). In addition, we determine the weak Signed Roman k-domatic number of some Graphs.

Let k ≥,1 be an integer, and let G be a finite and simple Graph with vertex set V (G). A Signed total Italian k-dominating function on a Graph G is a function f: V (G) →, {-1,1,2} such that ∑, u, N(v) f(u) ≥,k for every v ,V (G), where N(v) is the neighborhood of v, and each vertex u with f(u) =-1 is adjacent to a vertex v with f(v) = 2 or to two vertices w and z with f(w) = f(z) = 1. A set {f1,f2, …, ,fd} of distinct Signed total Italian k-dominating functions on G with the property that ∑, d i=1 fi(v) ≥,k for each v ,V (G), is called a Signed total Italian k-dominating family (of functions) on G. The maximum number of functions in a Signed total Italian k-dominating family on G is the Signed total Italian k-domatic number of G, denoted by dk stI (G). In this paper we initiate the study of Signed total Italian k-domatic numbers in Graphs, and we present sharp bounds for dk stI (G). In addition, we determine the Signed total Italian k-domatic number of some Graphs.

A Signed Graph (or, in short, siGraph) S = (Su, s) consists of an underlying Graph Su:= G = (V, E) and a function s: E (Su) ®{+, -}, called the signature of S. A marking of S is a function m: V (S) ®{+, -}. The canonical marking of a Signed Graph S, denoted ms, is given asms (v):= Õ vw2E(S) s (vw).The line Graph of a Graph G, denoted L (G), is the Graph in which edges of G are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in G. There are three notions of a line Signed Graph of a Signed Graph S = (Su, s) in the literature, viz., L (S), Lx (S) and L· (S), all of which have L (Su) as their underlying Graph, only the rule to assign signs to the edges of L (Su) differ. Every edge ee ′in L (S) is negative whenever both the adjacent edges e and e’ in S are negative, an edge ee′in Lx (S) has the product s (e) s (e′) as its sign and an edge ee′in L· (S) has ms (v) as its sign, where vÎ V (S) is a common vertex of edges e and e′.The line-cut Graph (or, in short, lict Graph) of a Graph G = (V,E), denoted by Lc (G), is the Graph with vertex set E (G) È C (G), where C (G) is the set of cut-vertices of G, in which two vertices are adjacent if and only if they correspond to adjacent edges of G or one vertex corresponds to an edge e of G and the other vertex corresponds to a cut-vertex c of G such that e is incident with c.In this paper, we introduce dot-lict Signed Graph (or ·-lict Signed Graph) L·c (S), which has Lc (Su) as its underlying Graph. Every edge uv in L·c (S) has the sign ms (p), if u, v Î E (S) and p Î V (S) is a common vertex of these edges, and it has the sign ms (v), if u Î E (S) and v Î C (S). We characterize Signed Graphs on Kp, p³2, on cycle Cn and on Km, n which are ·-lict Signed Graphs or ·-line Signed Graphs, characterize Signed Graphs S so that L·c (S) and L· (S) are balanced. We also establish the characterization of Signed Graphs S for which S ~ L·c (S), S ~ L· (S), h (S) ~ L·c (S) and h (S) ~ L· (S), here h (S) is negation of S and ~ stands for switching equivalence.

In this article, we study the Signed distance matrix of the product of Signed Graphs such as the Cartesian product and the lexicoGraphic product in terms of the Signed distance matrices of the factor Graphs. Also, we discuss the distance spectra of some special classes of product of Signed Graphs.

A Signed Graph is an ordered pair  = (G;  ); where G = (V; E) is the underlying Graph of  with a signature function  : E! f1; 1g. In this article, we de ne the n th power of a Signed Graph and discuss some properties of these powers of Signed Graphs. As we can de ne two types of Signed Graphs as the power of a Signed Graph, necessary and su cient conditions are given for an n power of a Signed Graph to be unique. Also, we characterize balanced power Signed Graphs.

The index λ, 1(ґ, ) of a Signed Graph ґ,= (G,σ, ) is just the largest eigenvalue of its adjacency matrix. For any n > 4 we identify the Signed Graphs achieving the minimum index in the class of Signed bicyclic Graphs with n vertices. Apart from the n = 4 case, such Graphs are obtained by considering a starlike tree with four branches of suitable length (i. e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all Signed bicyclic Graphs containing a theta-Graph and whose index is less than 2 are detected.