The main goal of this research article is to study the morphic property of the quotient ring of a skew (left) polynomial ring in [Formula: see text] by [Formula: see text]. We prove that if [Formula: see text] is an endomorphism of a ring [Formula: see text] and [Formula: see text] then [Formula: see text] is von Neumann regular if [Formula: see text] is left morphic in [Formula: see text]. Using this result, it is shown that a ring [Formula: see text] is unit regular if [Formula: see text] is left morphic. Further, we prove that for any endomorphism [Formula: see text] of a ring [Formula: see text], the ring [Formula: see text] is left morphic and [Formula: see text] is abelian if and only if [Formula: see text] is strongly regular and [Formula: see text] for all [Formula: see text]. Also it is shown that for any endomorphism [Formula: see text] of a ring [Formula: see text], the ring [Formula: see text] is left centrally morphic if and only if [Formula: see text] is strongly regular and there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text].

A pair of operators [Formula: see text] is said to be an [Formula: see text]-biisometric operator (or an [Formula: see text]-biisometry) if it satisfies the equation [Formula: see text] In this paper, we first present several properties associated with this family. Then, we prove that if [Formula: see text] is an [Formula: see text]-biisometry and [Formula: see text] are nilpotent operators with order [Formula: see text] and [Formula: see text] respectively, then [Formula: see text] is [Formula: see text]-biisometric operator where [Formula: see text]. Finally, we characterize the spectra of these operators.

We present a method for constructing harmonic immersions in [Formula: see text], known as the Enneper-type representation. We also prove that any harmonic immersion in [Formula: see text] can be obtained using this approach. Furthermore, we determine the number of non-planar rotational harmonic immersions in [Formula: see text] that connect two coaxial circles in parallel planes, where both circles have the same radius [Formula: see text] and are separated by a distance [Formula: see text].

Let [Formula: see text] and [Formula: see text] respectively denote the chromatic number and clique number of a graph [Formula: see text]. A [Formula: see text] is a path on 5 vertices, a banner (paw) is the graph obtained by joining a new vertex to a single vertex of [Formula: see text] ([Formula: see text]) and a hammer is obtained by subdividing the pendant edge of a paw exactly once. Recently, [Formula: see text]-free graphs have received wide attention. In 2019, Karthick, Maffray and Pastor, gave a structural characterisation of [Formula: see text]-free graphs, which when combined with a result by Bourneuf and Thomassé (2023) implies that for a [Formula: see text]-free graph [Formula: see text], [Formula: see text]. Geißer in his thesis (2022) showed that the [Formula: see text]-binding function of the class of [Formula: see text]-free graphs is bounded by the [Formula: see text]-binding function of [Formula: see text]-free graphs. By a result of Kim (1995), the chromatic number [Formula: see text] of a [Formula: see text]-free graph [Formula: see text] has order of magnitude [Formula: see text]. Recently, Song and Xu (2024) proved that every ([Formula: see text], [Formula: see text], banner, hammer)-free graph [Formula: see text] is [Formula: see text]-colorable. This motivates us to study the subclasses of ([Formula: see text], banner)-free graphs. We prove that for any ([Formula: see text], banner, [Formula: see text])-free graph [Formula: see text] where [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, the bound is tight for [Formula: see text].

The eccentricity matrix 𝔼(G) of a graph G is constructed from the distance matrix D(G) by only retaining the leading values in each row and each column and setting the rest of the elements in the corresponding row and column to zero. The article aims to study the concepts of energy and the inertia of the eccentricity matrices of graphs obtained by means of various graph products, like the strong product, the Cartesian product, and the Kronecker product, and well-established relations are derived in terms of their base graphs. Furthermore, we construct infinitely many pairs of non-isomorphic graphs having the same set of eigenvalues with respect to the eccentricity matrix.

This article is devoted to the investigation of α-Ricci and conformal α–Ricci solitons in Vaidya spacetime. The main result obtained is that the potential vector field of an α–Ricci soliton or conformal α–Ricci soliton in Vaidya spacetime is homothetic. Moreover, the explicit expressions for potential vector fields are also obtained. We also characterize the Vaidya spacetime admitting gradient α–Ricci or conformal α–Ricci solitons. It turns out that there does not exist any nonconstant generating potential function for gradient α–Ricci solitons as well as gradient conformal α–Ricci solitons. Finally, we completely characterize the space of all homothetic vector fields and in particular Killing vector fields in Vaidya spacetime.

‘A semigroup is completely regular if and only if it is a union of groups’- an analogue of this structure theorem of completely regular semigroup has been obtained in the setting of seminearrings in [[16], Mukherjee (Pal) et al., Semigroup Forum (2018)]. In it, a class of seminearrings (called generalized left completely regular seminearrings, abbreviated as GLCR) has been characterized as a union of near-rings. This work has been extended in the present article to characterize the seminearrings which are union of various types (regular, completely regular, inverse, Clifford) of regular near-rings.

Centrality measures are the tools used in most network analysis to rank the vertices of the network. Depending on different perspectives, many important centrality measures are defined and used for network analysis. Stress is one such important centrality measure of graphs applicable to the study of social and biological networks. Stress centrality evaluates the importance of a node by counting how many shortest paths pass through it. A node with higher stress centrality tends to be the one that plays a critical role in connecting different parts of the network, as it lies on a large number of shortest paths between other nodes. In this paper, we obtain the stress centrality of vertices in the product graphs with reference to the unary operations, namely, the splitting graph, the shadow graph, and the Mycielski graph. Further, we have given the expressions for the total stress of a few standard classes of graphs with respect to these operations.

In this paper, we introduce and investigate novel notions of generalized difference statistical convergence of order [Formula: see text] [Formula: see text] within the framework of neutrosophic [Formula: see text]-normed linear spaces. We develop and analyze the concepts of generalized difference statistical convergence, lacunary statistical convergence, and lacunary convergence of order [Formula: see text] via ideals. A significant result shows that strongly [Formula: see text]-[Formula: see text]-lacunary convergence of order [Formula: see text] implies [Formula: see text]-[Formula: see text]-lacunary statistical convergence of the same order, although the converse does not hold in general. However, this converse implication becomes true when [Formula: see text] and the sequence in question is [Formula: see text]-bounded. We further explore the inclusion relationships between generalized difference statistical convergence and lacunary statistical convergence of order [Formula: see text] with respect to ideals in the neutrosophic [Formula: see text]-normed context. Under the condition [Formula: see text], we establish hierarchical inclusion results for the set of all generalized difference [Formula: see text]-lacunary statistically convergent sequences. Additionally, we present further results on these sets based on a newly defined lacunary sequence satisfying the same order constraints. Some intriguing scenarios remain unresolved, and we conclude by posing these as open problems for future research within neutrosophic [Formula: see text]-normed linear spaces.

In this paper, we present the concept of induced topological pressure for random walks governed by finitely generated semigroup actions on compact metric spaces. Concentrating on the composition of endomorphisms under a unilateral shift, we build upon prior ergodic studies and derive a variational principle for induced pressure. This work provides fresh perspectives on the thermodynamic behavior of such systems.