# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### On the connected detour number of a graph

JPRM-Vol. 1 (2009), Issue 1, pp. 149 – 170 Open Access Full-Text PDF
A. P. Santhakumaran, S. Athisayanathan
Abstract: For two vertices u and v in a graph $$G = (V, E)$$, the detour distance $$D(u, v)$$ is the length of a longest $$u–v$$ path in $$G$$. A $$u–v$$ path of length $$D(u, v)$$ is called a $$u–v$$ detour. A set $$S ⊆ V$$ is called a detour set of $$G$$ if every vertex in $$G$$ lies on a detour joining a pair of vertices of $$S$$. The detour number $$dn(G)$$ of G is the minimum order of its detour sets and any detour set of order $$dn(G)$$ is a detour basis of $$G$$. A set $$S ⊆ V$$ is called a connected detour set of $$G$$ if S is detour set of $$G$$ and the subgraph $$G[S]$$ induced by S is connected. The connected detour number $$cdn(G)$$ of $$G$$ is the minimum order of its connected detour sets and any connected detour set of order $$cdn(G)$$ is called a connected detour basis of $$G$$. Graphs G with detour diameter $$D ≤ 4$$ are characterized when $$cdn(G) = p$$, $$cdn(G) = p−1$$, $$cdn(G) = p−2$$ or $$cdn(G) = 2$$. A subset $$T$$ of a connected detour basis $$S$$ of $$G$$ is a forcing subset for $$S$$ if $$S$$ is the unique connected detour basis containing $$T$$. The forcing connected detour number $$fcdn(S)$$ of $$S$$ is the minimum cardinality of a forcing subset for $$S$$. The forcing connected detour number $$fcdn(G)$$ of $$G$$ is $$min{fcdn(S)}$$, where the minimum is taken over all connected detour bases $$S$$ in $$G$$. The forcing connected detour numbers of certain classes of graphs are determined. It is also shown that for each pair $$a$$, $$b$$ of integers with $$0 ≤ a < b$$ and $$b ≥ 3$$, there is a connected graph $$G$$ with $$fcdn(G) = a$$ and $$cdn(G) = b$$.

### Exact solutions of generalized Oldroyd-B fluid subject to a time-dependent shear stress in a pipe

JPRM-Vol. 1 (2009), Issue 1, pp. 139 – 148 Open Access Full-Text PDF
Qammar Rubbab, Syed Muhammad Husnine, Amir Mahmood
Abstract: The velocity field and the shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder subject to a longitudinal time-dependent shear stress are determined by means of Hankel and Laplace transforms. The exact solutions, written in terms of the generalized $$G$$-functions, satisfy all imposed initial and boundary conditions. The similar solutions for ordinary Oldroyd-B, ordinary and generalized Maxwell, ordinary and generalized second grade as well as for Newtonian fluids are obtained as limiting cases of our general solutions.

### On the Ramsey number for paths and beaded wheels

JPRM-Vol. 1 (2009), Issue 1, pp. 133 – 138 Open Access Full-Text PDF
Kashif Ali, Edy Tri Baskoro, Ioan Tomescu
Abstract: For given graphs $$G$$ and $$H$$, the Ramsey number $$R(G, H)$$ is the least natural number n such that for every graph $$F$$ of order $$n$$ the following condition holds: either $$F$$ contains $$G$$ or the complement of $$F$$ contains $$H$$. Beaded wheel $$BW_{2,m}$$ is a graph of order $$2m + 1$$ which is obtained by inserting a new vertex in each spoke of the wheel $$W_m$$. In this paper, we determine the Ramsey number of paths versus Beaded wheels: $$R(P_n, BW_{2,m}) = 2n − 1$$ or $$2n$$ if $$m ≥ 3$$ is even or odd, respectively, provided $$n ≥ 2m^2 − 5m + 4$$.

### HYBRID FUNCTIONS APPROACH FOR SOLVING FREDHOLM AND VOLTERRA INTEGRAL EQUATIONS

JPRM-Vol. 1 (2009), Issue 1, pp. 124 – 132 Open Access Full-Text PDF
T. Shojaeizadeh, Z. Abadi, E. Golpar Raboky
Abstract: This paper presents a computational technique for Fredholm and Volterra integral equations of the second kind. The method based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and legendre polynomials are presented. The operational matrices of integration and product are utilized to reduce the computation of integral equation into some algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

### $$C^2$$ Rational quintic function

JPRM-Vol. 1 (2009), Issue 1, pp. 115 – 126 Open Access Full-Text PDF
Maria Hussain, Malik Zawwar Hussain, Robert J. Cripps
Abstract: A two-parameter family of piecewise $$C^2$$ rational quintic functions is presented along with an error investigation for the approximation of an arbitrary $$C^3$$ function. The two parameters have a direct geometric interpretation making their use straightforward. Illustrations of their effect on the shape of the rational function are given. The relaxed continuity constraints and the increased flexibility via the two parameters make the
proposed function a suitable candidate for interactive CAD.

### The connected vertex geodomination number of a graph

JPRM-Vol. 1 (2009), Issue 1, pp. 101 – 114 Open Access Full-Text PDF
A. P. Santhakumaran, P. Titus
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 $$G$$.