Search

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

We show that, up to isomorphism, there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n ≥ 4. This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs (Discrete Math., 256, 301–334 (2002)). As an application, a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups, which generalises the earlier formula of Huang et al. dealing with the particular case when n is a prime (Acta Math. Sin., Engl. Ser., 33, 996–1011 (2017)). As another application, a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve (arXiv preprint, (2007)). [ABSTRACT FROM AUTHOR]

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds M = G / H such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition g = h ⊕ n (vector space direct sum) with n nilpotent. When the metric is nondegenerate on [ n , n ] , we show that n is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on [ n , n ] , we show that n is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded. [ABSTRACT FROM AUTHOR]