Journal of Prime Research in Mathematics

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 \(d_2 = a\), \(D_2 = b\).

Abstract: We employ the techniques of mixed Hodge modules in order to answer some questions on extension of mixed Hodge structures. Specifically a theorem of M. Saito tells that, the mixed Hodge modules on a complex algebraic manifold X, correspond to polarized variation of mixed Hodge structures on Zariski open dense subsets of X. In this article we concern with the minimal extension of MHM or PVMHS related to this criteria. In [26] we studied the extension of VMHS associated to isolated hypersurface singularities. This article generalizes some of the results there to the admissible VMHS on open dense submanifolds. Some applications to the Neron models of Hodge structures are also given. A short discussion on abelian positivity in the positive characteristic and of height pairing on arithmetic varieties have been included.

Abstract: Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal polynomials of low degree. In this paper, we consider a recurrence relation that has not been studied before and which involves a relatively higher degree polynomial. Interestingly, it leads to an algorithm that shows superior stability when compared to existing Lanczos-type algorithms. This new algorithm is derived and described. It is then compared to the best known algorithms of this type, namely \(A_5/B_{10}\), \(A_8/B_{10}\), as well as Arnoldi’s algorithm, on a set of standard test problems. Numerical results are included.