Journal of Prime Research in Mathematics

Abstract: Interval-valued fuzzy set theory is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, the concepts of interval-valued fuzzy (prime, semiprime) ideal and the Cartesian product of interval-valued fuzzy subsets have been introduced. Some interesting results about Cartesian product of interval-valued fuzzy ideals, interval-valued fuzzy prime ideals, intervalvalued fuzzy semiprime ideals, interval-valued fuzzy bi-ideals and intervalvalued fuzzy interior ideals in ordered semigroups are obtained. The purport of this paper is to link ordinary ideals with interval-valued fuzzy ideals by means of level subset of Cartesian product of interval-valued fuzzy subsets.

Abstract: In this article, a new alteration of the Homotopy Perturbation Method (HPM) is proposed to approximate the solution of unsteady axisymmetric flow of Newtonian fluid. The flow is squeezed between two circular plates and passes through a porous medium channel. The alteration extends the Homotopy Perturbation with a Laplace transform, which is referred to as the Laplace Transform Homotopy Perturbation Method (LTHPM) in this manuscript. A single fourth order non-linear ordinary differential equation is obtained using similarity transformations. The resulting boundary value problem is then solved through LTHPM, HPM and fourth order Implicit Runge Kutta Method (IRK4). Convergence of the proposed scheme is checked by finding absolute residual errors of various order solutions. Also, the validity is confirmed by comparing numerical and analytical (LTHPM) solutions. The comparison of obtained residual errors shows that LTHPM is an effective scheme that can be applied to various initial and boundary value problems in science and engineering

Abstract: In this paper, the exact formula for the reciprocal product degree distance of strong product of a connected graph and the complete multipartite graph with partite sets of sizes \(m_0, m_1, . . . , m_{r−1}\) is obtained. Using the results obtained here, the formula for the reciprocal degree distance of the closed fence graph is computed.

Abstract: The aim of this article is to give Hardy-type inequalities involving linear differential operator, Widder’s derivative and generalized fractional integral using superquadratic functions.

Abstract: Let \(v\) be a vertex and \(C\) a clique in a connected graph \(G\). A vertex-to-clique \(u − C\) path P is a \(u − v\) path, where v is a vertex in \(C\) such that \(P\) contains no vertices of \(C\) other than \(v\). The vertex-to-clique distance, \(d(u, C)\) is the length of a smallest \(u−C\) path in \(G\). A \(u−C\) path of length \(d(u, C)\) is called a \(u − C\) geodesic. The vertex-to-clique eccentricity \(e_1(u)\) of a vertex \(u\) in \(G\) is the maximum vertex-to-clique distance from \(u\) to a clique \(C ∈ ζ\), where \(ζ\) is the set of all cliques in \(G\). The vertex-to-clique radius \(r_1\) of \(G\) is the minimum vertex-to-clique eccentricity among the vertices of \(G\), while the vertex-to-clique diameter \(d_1\) of \(G\) is the maximum vertex-to-clique eccentricity among the vertices of \(G\). Also the vertex toclique detour distance, \(D(u, C)\) is the length of a longest \(u−C\) path in \(G\). A \(u−C\) path of length \(D(u, C)\) is called a \(u−C\) detour. The vertex-to-clique detour eccentricity \(e_{D1}(u)\) of a vertex \(u\) in \(G\) is the maximum vertex-toclique detour distance from u to a clique \(C ∈ ζ\) in \(G\). The vertex-to-clique detour radius \(R_1\) of \(G\) is the minimum vertex-to-clique detour eccentricity among the vertices of \(G\), while the vertex-to-clique detour diameter \(D_1\) of \(G\) is the maximum vertex-to-clique detour eccentricity among the vertices of \(G\). It is shown that \(R_1 ≤ D_1\) for every connected graph \(G\) and that every two positive integers a and b with \(2 ≤ a ≤ b\) are realizable as the vertex-to-clique detour radius and the vertex-to-clique detour diameter, respectively, of some connected graph. Also it is shown that for any three positive integers \(a\), \(b\), \(c\) with \(2 ≤ a ≤ b < c\), there exists a connected graph G such that \(r_1 = a\), \(R_1 = b\), \(R = c\) and for any three positive integers \(a\), \(b\), \(c\) with \(2 ≤ a ≤ b < c\) and \(a + c ≤ 2b\), there exists a connected graph \(G\) such that \(d_1 = a\), \(D_1 = b\), \(D = c\).

Abstract: Let \(G\) be a graph and some pebbles are distributed on its vertices. 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 of a graph \(G\) is the least integer \(m\) such that from any distribution of m pebbles on the vertices of \(G\), we can move t pebbles to any specified vertex by a sequence of pebbling moves. In this paper, we determine the t-pebbling number of some wheel related graphs.