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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 the 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 properties of the Cayley sum signed graph. [ABSTRACT FROM AUTHOR]

The concept of associating a graph to a ring was initiated by Beck in 1988. The zero-divisor graph of a commutative ring R with unity (1 ≠ 0) is defined as a simple graph, denoted by Γ (R) , where elements of the ring R represent vertices and two distinct vertices in Γ (R) are adjacent if the product of the corresponding elements is zero in R. 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. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the edges incident with the vertex in e ∩ e′. In this paper we establish a structural characterization of •-line sigraphs, extending a well known characterization of line graphs due to Harary. Further we study several standard properties of •-line sigraphs, such as the balanced •-line sigraphs, sign-compatible •-line sigraphs and C-sign-compatible •-line sigraphs.

A s i g n e d g r a p h is a graph whose edges carry the weight '+' or '−'. A signed graph S is called sign-regular if d − (v) is same for all v ∈ V and d + (v) is same for all v ∈ V. The problems of embedding (i , j) -sign-regular signed graphs in (i + k , j + l) -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. [ABSTRACT FROM AUTHOR]