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The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal of a ring is called an annihilating ideal if there exists such that . As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727-739], for any ring , we denote by the set of all annihilating ideals of and by the set of all nonzero annihilating ideals of . Let be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by , which is an undirected simple graph whose vertex set is and distinct elements are joined by an edge in this graph if and only if . The aim of this paper is to study the interplay between the ring theoretic properties of a ring and the graph theoretic properties of , where is the complement of . In this paper, we first determine when is connected and also determine its diameter when it is connected. We next discuss the girth of and study regarding the cliques of . Moreover, it is shown that is complemented if and only if is reduced. [ABSTRACT FROM AUTHOR]

The Kautz digraphs K(d, ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related to these, the cyclic Kautz digraphs CK(d, ℓ) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs sK(d, ℓ), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d, ℓ) and CK(d, ℓ). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d, ℓ) and CK(d, ℓ) are maximally vertex-connected and super-edge-connected. Whereas K(d, ℓ) are optimal with respect to the diameter, we show that sK(d, ℓ) and CK(d, ℓ) are optimal with respect to the mean distance, whose exact values are given for both families when ℓ = 3. Finally, we provide a lower bound on the girth of CK(d, ℓ) and sK(d, ℓ). [ABSTRACT FROM AUTHOR]