Journal of Prime Research in Mathematics

Abstract: This paper presents the heat transfer of a third grade fluid between two heated parallel plates for two models: constant viscosity model and Reynold’s model. In both cases the nonlinear energy and momentum equations have been solved by HAM. The graphs for the velocity and temperature profiles are plotted and discussed for various values of the emerging parameters in the problem. The main effect is governed by whether or not the fluid is non-Newtonian and the temperature effects are being referred to have a less dominant role.

Abstract: In this paper we introduce the notion of \((λ, µ)\)-Fuzzy ternary subsemirings and \((λ, µ)\)-Fuzzy ideals in ternary semirings which can be regarded as the generalization of fuzzy ternary subsemirings and fuzzy ideals in ternary semirings.

Abstract: We are investigating the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: The multiplication group of an AG-group of order n is a non-abelian group of order 2n and its left section is an abelian group of order n. The inner mapping group of an AG-group of any order is a cyclic group of order 2.

Abstract: Let \(C\) be a configuration of pebbles on a graph \(G\). A pebbling move (step) consists of removing two pebbles from one vertex, throwing one pebble away, and moving the other pebble to an adjacent vertex. The \(t\)-pebbling number, \(f_t(G)\), of a connected graph \(G\), is the smallest positive integer such that from every configuration of \(f_t(G)\) pebbles, t pebbles can be moved to any specified target vertex by a sequence of pebbling moves. In this paper, we determine the t-pebbling number for squares of cycles.

Abstract: Let \(G\) be a finite group and \(H\) a subgroup of \(G\). In 2012, David F. Anderson et al. introduced the subgroup graph of \(H\) in \(G\) as a simple graph with vertex set consisting all elements of G and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy ∈ H\). They denoted this graphby \(Γ_H(G)\). In this paper, we consider the complement of \(Γ_H(G)\), denoted by \(\overline{Γ_H(G)}\) and will give some graph properties of this graph, for instance diameter, girth, independent and dominating sets, regularity. Moreover, the structure of this graph, planerity and 1-planerity are also investigated in the paper.

Abstract: Let \(C\) be a clique and \(v\) a vertex in a connected graph \(G\). A clique-to-vertex \(C − v\) path \(P\) is a \(u − v\) path, where u is a vertex in \(C\) such that \(P\) contains no vertices of \(C\) other than \(u\). The clique-to-vertex distance, \(d(C, v)\) is the length of a smallest \(C − v\) path in \(G\). A \(C − v\) path of length \(d(C, v)\) is called a \(C − v\) geodesic. The clique-to-vertex eccentricity \(e_2(C)\) of a clique \(C\) in G is the maximum clique-to-vertex distance from \(C\) to a vertex \(v ∈ V\) in \(G\). The clique-to-vertex radius \(r_2\) of \(G\) is the minimum clique-to-vertex eccentricity among the cliques of \(G\), while the clique-to-vertex diameter \(d-2\) of \(G\) is the maximum cliqueto-vertex eccentricity among the cliques of \(G\). Also The clique-to-vertex detour distance, \(D(C, v)\) is the length of a longest \(C − v\) path in \(G\). A \(C −v\) path of length \(D(C, v)\) is called a  (C −v\) detour. The clique-to-vertex detour eccentricity \(e_{D2}(C)\) of a clique \(C\) in \(G\) is the maximum clique-tovertex detour distance from \(C\) to a vertex \(v ∈ V\) in  (G\). The clique-to-vertex detour radius \(R_2\) of \(G\) is the minimum clique-to-vertex detour eccentricity among the cliques of \(G\), while the clique-to-vertex detour diameter \(D_2\) of \(G\) is the maximum clique-to-vertex detour eccentricity among the cliques of \(G\). It is shown that \(R_2 ≤ D_2\) for every connected graph \(G\) and that every two positive integers a and b with \(2 ≤ a ≤ b\) are realizable as the clique-tovertex detour radius and the clique-to-vertex detour diameter respectively of some connected graph. Also it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(r_2 = a\), \(R_2 = b\) and it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(d_2 = a\), \(D_2 = b\).