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Let ℤn be the ring of integer modulo n with two binary operators, addition (+) and multiplication (.), where n is a positive integer. The special set 풮 is defined as 풮 = {a ∈ ℤn : (∃b ∈ ℤn) ba ≡ a, b≢a, b≢1}. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set 풮 and denote it by Ω−(Z n). In this paper, we define a relationship between G and Ge∗, Ge∗ is a derived graph from G by removing r edges, where r is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and Ω−(Z n). We also provide values of n for which the graph Ω−(ℤ n) is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness. [ABSTRACT FROM AUTHOR]

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]

Formally, a signed graphS is a pair () that consists of a graph and a sign mapping called signature from E to the sign group {+, -}. Given a signed graph S and a positive integer t, the t-path signed graph of S is a signed graph whose vertex set is V(S) and two vertices are adjacent if and only if there exists a path of length t between these vertices and then by defining its sign to be '-' if and only if in every such path of length t in S all the edges are negative. The negation of a signed graph S is a signed graph obtained from S by reversing the sign of every edge of S. Two signed graphs and on the same underlying graph are switching equivalent if it is possible to assign signs ' ' ('plus') or ' ' ('minus') to the vertices of such that by reversing the sign of each of its edges that have received opposite signs at its ends, one obtains. In this paper, we characterize signed graphs whose negations are switching equivalent to their t-path signed graphs for and also characterize signed graphs such that the spectrum of their t-path signed graphs, where , and 2, is symmetric about the origin. [ABSTRACT FROM AUTHOR]