Journal of Prime Research in Mathematics

Abstract: For a connected graph \(G\) of order \(p ≥ 2\), a set \(S ⊆ V (G)\) is an \(x\)-geodominating set of \(G\) if each vertex \(v ∈ V (G)\) lies on an \(x-y\) geodesic for some element y in \(S\). The minimum cardinality of an \(x\)-geodominating set of G is defined as the \(x\)-geodomination number of \(G\), denoted by gx(G). An \(x\)-geodominating set of cardinality \(g_x(G)\) is called a \(g_x\)-set of \(G\). A connected \(x\)-geodominating set of G is an \(x\)-geodominating set S such that the subgraph \(G[S]\) induced by \(S\) is connected. The minimum cardinality of a connected \(x\)-geodominating set of \(G\) is defined as the connected \(x\)-geodomination number of \(G\) and is denoted by \(cg_x(G)\). A connected \(x\)-geodominating set of cardinality \(cg_x(G)\) is called a \(cg_x\)-set of \(G\). We determine bounds for it and find the same for some special classes of graphs. If \(p, a\) and \(b\) are positive integers such that \(2 ≤ a ≤ b ≤ p − 1\), then there exists a connected graph G of order \(p\), \(g_x(G) = a\) and \(cg_x(G) = b\) for some vertex \(x\) in \(G\). Also, if \(p\), \(d\) and \(n\) are integers such that \(2 ≤ d ≤ p − 2\) and \(1 ≤ n ≤ p\), then there exists a connected graph \(G\) of order \(p\), diameter \(d\) and \(cg_x(G) = n\) for some vertex \(x\) in \(G\). For positive integers \(r\), \(d\) and \(n\) with \(r ≤ d ≤ 2r\), there exists a connected graph \(G\) with rad \(G = r\), \(diam G = d\) and \(cg_x(G) = n\) for some vertex \(x\) in \(G\).

Abstract: Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe more realistic features of the phenomenon. Main directions of the generalization of Richardson’s model are: 1) the dependence of platelet activation time on the distance from the injured vessel wall; 2) the nonhomogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall. The generalization of the model corresponds to the main experimental results concerning thrombi formation obtained in recent years. The extended model permits to achieve a numerical agreement between model and experimental data.

Abstract: In this paper we develop the theory of almost periodic functions defined on \(\mathbb{R}^n\) with values in locally convex spaces and Frechet spaces.

Abstract: The velocity field and the adequate shear stress, corresponding to the unsteady flow of generalized second grade fluids due to a constantly accelerating circular cylinder, are determined by means of the Hankel and Laplace transforms. The solutions that have been obtained satisfy all imposed initial and boundary conditions and for \(β → 1\) reduce to the similar solutions for the second grade fluids performing the same motion.

Abstract: This paper presents an algorithm for computing Longest Common Subsequences for two sequences. Given two strings \(X\) and \(Y\) of length \(m\) and \(n\), we present a greedy algorithm, which requires \(O(n log s)\) preprocessing time, where s is distinct symbols appearing in string \(Y\) and \(O(m)\) time to determines Longest Common Subsequences.

Abstract: Let G be a connected graph. For a vertex \(v ∈ V (G)\) and an ordered \(k-\)partition \(Π = {S_1, S_2, …, S_k}\) of \(V (G)\), the representation of \(v\) with respect to \(Π\) is the \(k-\)vector r \((v|Π) = (d(v, S_1), d(v, S_2), …, d(v, S_k))\) where \(d(v, S_i) = min_{w∈S_i} d(v, w)(1 ≤ i ≤ k)\). The k-partition \(Π\) is said to be resolving if the k-vectors \(r(v|Π), v ∈ V (G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V (G)\) is called the partition dimension of \(G\), denoted by \(pd(G)\). In this paper, we give upper bounds for the cardinality of vertices in some wheel related graphs namely gear graph, helm, sunflower and friendship graph with given partition dimension \(k\).