Jul 2026· Journal of Algebra and its Applications· 0 citations
Abstract
Consider a group [Formula: see text] and construct its power graph, whose vertex set consists of the elements of [Formula: see text]. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group [Formula: see text], the dihedral group [Formula: see text], and the dicyclic group [Formula: see text]. For [Formula: see text] the power graph of the cyclic group [Formula: see text] is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups
For a given positive integer [Formula: see text], the prime-[Formula: see text] graph denoted by [Formula: see text] is defined on the vertex set comprising all positive divisors of [Formula: see text] greater than 1. An edge exists between two distinct vertices [Formula: see text] and [Formula: see text] if and only if their greatest common divisor, [Formula: see text], is a prime factor of [Formula: see text]. This study explores the fundamental structural characteristics of [Formula: see text] systematically. Key findings establish that the graph is always connected for any integer [Formula: see text], with a diameter of at most [Formula: see text] and a radius of [Formula: see text]. The paper provides characterizations and formulas for various graph invariants, including the clique number, chromatic number, girth, and vertex degrees, demonstrating their direct dependence on the prime factorization of [Formula: see text]. It is shown that graphs [Formula: see text] and [Formula: see text] are isomorphic if [Formula: see text] and [Formula: see text] share the same prime factorization structure irrespective of the prime factors. Furthermore, conditions for planarity are determined. The analysis also covers properties such as the independence number, covering number, density of a graph establishing a relation between them and prime factorization of [Formula: see text]. This research illuminates the deep interplay between the arithmetic properties of integers and the resulting topological features of their associated graphs.
Graphs whose edge rings have regularity two are not completely characterized. We, therefore, study the minimal graded free resolution of edge rings of certain families of graphs, such as the general barbell graph [Formula: see text] (or [Formula: see text]-barbell graph) and the complete [Formula: see text]-sunlet graph [Formula: see text], and prove that the edge ring of a barbell graph is of regularity two. We give combinatorial formulas for the graded Betti numbers in the linear strand of Stanley-Reisner rings of [Formula: see text] and [Formula: see text]. We also determine the Hilbert series of the edge ring of [Formula: see text]-barbell graph in terms of [Formula: see text]-faces of its independence complex and in terms of graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]. Then we compute other graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]-complexes.
Shahnawaz Ahmad Rather, S. Pirzada· Algebra Colloquium· 0 citations
Let [Formula: see text] be a graph with no isolated vertex. A dominated coloring of [Formula: see text] is a proper coloring of [Formula: see text] such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of [Formula: see text] is called the dominated chromatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we study the dominated chromatic number of central graphs. We obtain some tight bounds for the dominated chromatic number of a central graph [Formula: see text] in terms of some invariants of the graph [Formula: see text]. Also we characterize the dominated chromatic number of the central graph of some families of graphs such as star graphs, path graphs, spider graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the dominated chromatic number of central graphs.
Let [Formula: see text] be a commutative ring, and let [Formula: see text] denote the set of all nonzero elements of [Formula: see text]. The weak zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph whose vertex set consists of the elements in [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if there exists a positive integer [Formula: see text] such that [Formula: see text].
In this study, we investigate finite commutative rings [Formula: see text] whose associated weak zero-divisor graphs [Formula: see text] belong to certain well-established classes of graphs. Specifically, we provide a classification of finite rings [Formula: see text] for which the graph [Formula: see text] is a unicyclic graph, a tree, a split graph, a planar graph, or an outerplanar graph. In addition, we explore the conditions under which [Formula: see text] is a toroidal graph.
Sandip Rawade, M. Nazim, C. Abdioglu et al.· Asian-European Journal of Ma...· 0 citations
The eccentricity of any vertex [Formula: see text] in a connected graph [Formula: see text] is the length of the largest distance from [Formula: see text] to any other vertex in [Formula: see text]. The eccentric graph of any graph [Formula: see text], denoted by [Formula: see text], is a graph with the same vertex set as [Formula: see text] and two vertices in [Formula: see text] are adjacent if the distance between those vertices is equal to the eccentricity of either of the vertices. A graph [Formula: see text] is properly connected if there is a properly colored path between every pair of vertices in it. In this paper, we define the proper eccentric graph for any properly connected graph [Formula: see text]. We examine the connectivity, proper eccentricity, proper diameter, and several other graph invariants of a proper eccentric graph of the join of any two connected graphs.
Unknown authors· Journal of Interconnection N...· 0 citations
Let [Formula: see text] be a ring with unity. The idempotent graph [Formula: see text] of a ring [Formula: see text] is an undirected simple graph whose vertex set is [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is an idempotent element of [Formula: see text]. Razaghi et al. (A graph with respect to idempotents of a ring. J. Algebra Appl., 20(6):2150105, 8, 2021) studied basic properties of [Formula: see text] such as connectedness, diameter and girth. In this article, first we correct a structural result obtained by Razaghi et al. and determine the precise structure of the idempotent graph of local rings. Further, we obtain a necessary and sufficient condition on the ring [Formula: see text] such that [Formula: see text] is planar. We prove that [Formula: see text] is an outerplanar graph if and only if [Formula: see text] is a local ring. Moreover, we classify all the finite commutative rings [Formula: see text] such that [Formula: see text] is claw-free, cograph, split graph and threshold graph, respectively. We conclude that for a finite non-local commutative ring, the latter two graph classes of [Formula: see text] are equivalent if and only if [Formula: see text] is a Boolean ring.
Praveen Mathil, Jitender Kumar, Barkha Baloda· Journal of Algebra and its A...· 0 citations
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