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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]