Journal of Prime Research in Mathematics

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.

Abstract: In this paper we construct the topological structure of linear 2-normed space. This enable us to define the concept of open sets in linear 2-normed space and derive an analogue of Baire’s theorem and Banach Steinhaus theorem in linear 2-normed spaces.

Abstract: For a (molecular) graph, the first Zagreb index \(M_1\) is equal to the sum of squares of the degrees of vertices, and the second Zagreb index \(M_2\) is equal to the sum of the products of the degrees of pairs of adjacent vertices. Similarly, the first and second Zagreb coindices are defined as \(\overline{M}_1(G) = \sum_{ uv \notin E(G)} (d_G(u) + d_G(v))\) and \(\overline{M}_2(G)=\sum_{ uv \notin E(G)} d_G(u)d_G(v)\). In this paper, we compute the Zagreb indices and coindices of strong, tensor and edge corona product of two connected graphs. We apply some of our results to compute the Zagreb indices and coindices of open and closed fence graphs.

Abstract: The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy perturbation method gives the same series solution as in Adomian Decomposition Method and we get the exact solution using two iterations only.