Let R be a ring with identity. The nil-clean graph of R is a graph, denoted by GNC(R), whose vertex-set is the set R, and where two distinct vertices x and y are adjacent if and only if x + y is a nil-clean element of R. An element r ∈ R is called a nil-clean element if it can be decomposed a sum of an idempotent and a nilpotent element of R. Let G be a finite undirected graph. An automorphism φ of G is a permutation on the vertex-set V(G) such that the graph preserves adjacency, that is, φ(v1) is adjacent to φ(v2) if and only if v1 is adjacent to v2. The set of all automorphisms of G together with the composition operation of permutations forms the automorphism group of G. In this paper, we firstly compute the order of the automorphism groups of nil-clean graphs for the ring ℤn. And then we determine the structure of the automorphism groups of GNC(ℤn) for n = pk, pq, 2kpl, where p, q are distinct primes and k, l are positive integers.
A subfamily of Dyck words called tight Dyck words is seen to correspond, via a “castling” procedure, to the vertex set of an ordered tree T. From T, a “blowing” operation recreates the whole family ol Dyck words. The vertices of T can be elementarily updated all along T. This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. Mütze et al. This takes into account that the Dyck words represent: (a) the cyclic and dihedral vertex classes of odd and middle-levels graphs, respectively, and (b) the cycles of their 2-factors, as found by T. Mütze et al.
For any real number α, the general energy of a graph is defined as the sum of the α-th powers of the nonzero singular values of its adjacency matrix. This definition unifies several classical spectral invariants, such as the graph energy and spectral moments. In this paper, we establish bounds on the general energy of graphs. These bounds, in turn, yield new estimates for the ordinary energy and spectral moments, and lead to a more general relationship between these quantities.
