If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following:

Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(D_n, {ab^r, ab^s, ab^t}) is a GRR of D_n.

We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.

For a set \( S \) of vertices in a connected graph \( G \), the multiplicative distance of a vertex \( v \) with respect to \( S \) is defined by \(d_{S}^{*}(v) = \prod\limits_{x \in S, x \neq v} d(v,x).\) If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G \), then \( S \) is called a multiplicative distance-locating set of \( G \). The minimum cardinality of a multiplicative distance-locating set of \( G \) is called its multiplicative distance-location number \( loc_{d}^{*}(G) \). If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G-S \), then \( S \) is called an external multiplicative distance-locating set of \( G \). The minimum cardinality of an external multiplicative distance-locating set of \( G \) is called its external multiplicative location number \( loc_{e}^{*}(G) \). We prove the existence or non-existence of multiplicative distance-locating sets in some well-known classes of connected graphs. Also, we introduce a family of connected graphs such that \( loc_{d}^{*}(G) \) exists. Moreover, there are infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) exists as well as infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) does not exist. A lower bound for the multiplicative distance-location number of a connected graph is established in terms of its order and diameter.

Earlier optimal key pre\(-\)distribution schemes (KPSs) for distributed sensor networks (DSNs) were proposed using combinatorial designs via transversal designs, affine, and partially affine resolvable designs. Here, nearly optimal KPSs are introduced and a class of such KPSs is obtained from resolvable group divisible designs. These KPSs are nearly optimal in the sense of local connectivity. A metric for the efficiency of KPSs is given. Further, an optimal KPS has also been proposed using affine resolvable \( L_{2} \)-type design.

We study the nonzero algebraic real algebras \( A \) with no nonzero joint divisor of zero. We prove that if \( Z(A) \neq 0 \) and \( A \) satisfies one of the Moufang identities, then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). We show also that if \( A \) is power-associative, flexible, and satisfies the identity \( (a,a,[a,b])=0 \), then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). Finally, we prove that \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), and \( \mathbb{O} \) are the only algebraic real algebras with no nonzero divisor of zero satisfying the middle Moufang identity, or the right and left Moufang identities.

The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs \( G \) and \( H \) is the graph \( G \odot H \) obtained by taking one copy of \( G \), called the center graph, \( |V(G)| \) copies of \( H \), called the outer graph, and making the \( j^{th} \) vertex of \( G \) adjacent to every vertex of the \( j^{th} \) copy of \( H \), where \( 1 \leqslant j \leqslant |V(G)| \). The Join graph \( G + H \) of two graphs \( G \) and \( H \) is the graph with vertex set \( V(G + H)=V(G) \cup V(H) \) and edge set \( E(G + H)=E(G) \cup E(H) \cup \{uv :u \in V(G),v \in V(H)\} \). In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.