# 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

### Forcing edge detour number of an edge detour graph

JPRM-Vol. 1 (2010), Issue 1, pp. 13 – 21 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 an edge detour set if every edge in $$G$$ lies on a detour joining a pair of vertices of $$S$$. The edge detour number $$dn_1(G)$$ of G is the minimum order of its edge detour sets and any edge detour set of order $$dn_1(G)$$ is an edge detour basis of $$G$$. A connected graph $$G$$ is called an edge detour graph if it has an edge detour set. A subset $$T$$ of an edge detour basis $$S$$ of an edge detour graph $$G$$ is called a forcing subset for $$S$$ if $$S$$ is the unique edge detour basis containing $$T$$. A forcing subset for $$S$$ of minimum cardinality is a minimum forcing subset of $$S$$. The forcing edge detour number $$fdn_1(S)$$ of $$S$$, is the minimum cardinality of a forcing subset for $$S$$. The forcing edge detour number $$fdn_1(G)$$ of $$G$$, is $$min{fdn_1(S)}$$, where the minimum is taken over all edge detour bases $$S$$ in $$G$$. The general properties satisfied by these forcing subsets are discussed and the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters $$dn_1(G)$$ and $$fdn_1(G)$$ satisfy the relation $$0 ≤ fdn_1(G) ≤ dn_1(G)$$ and it is proved that for each pair $$a$$, $$b$$ of integers with $$0 ≤ a ≤ b$$ and $$b ≥ 2$$, there is an edge detour graph $$G$$ with $$fdn_1(G) = a$$ and $$dn_1(G) = b$$.

### Divisor path decomposition number of a graph

JPRM-Vol. 1 (2010), Issue 1, pp. 01 – 12 Open Access Full-Text PDF
K. Nagarajan, A. Nagarajan
Abstract: A decomposition of a graph G is a collection Ψ of edge-disjoint subgraphs $$H_1,H_2, . . . , H_n$$ of $$G$$ such that every edge of $$G$$ belongs to exactly one $$H_i$$. If each $$H_i$$ is a path in $$G$$, then $$Ψ$$ is called a path partition or path cover or path decomposition of $$G$$. A divisor path decomposition of a $$(p, q)$$ graph $$G$$ is a path cover $$Ψ$$ of $$G$$ such that the length of all the paths in $$Ψ$$ divides $$q$$. The minimum cardinality of a divisor path decomposition of $$G$$ is called the divisor path decomposition number of $$G$$ and is denoted by $$π_D(G)$$. In this paper, we initiate a study of the parameter $$π_D$$ and determine the value of $$π_D$$ for some standard graphs. Further, we obtain some bounds for $$π_D$$ and characterize graphs attaining the bounds.

### Exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders

JPRM-Vol. 1 (2009), Issue 1, pp. 192 – 204 Open Access Full-Text PDF
Amir Mahmood, Najeeb Alam Khan, Corina Fetecau, Muhammad Jamil, Qammar Rubbab
Abstract: The velocity field and associated shear stress corresponding to the longitudinal oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. The flow is due to both of the cylinders that at $$t = 0^+$$ suddenly begin to oscillate along their common axis with different angular frequencies of their respective velocities. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of Newtonian fluid are also obtained as limiting case. The flows of second grade and Newtonian fluids are compared graphically by plotting their velocity profiles.

### On F-derivations of BCI-algebras

JPRM-Vol. 1 (2009), Issue 1, pp. 171 – 191 Open Access Full-Text PDF
Farhat Nisar
Abstract: In this paper we introduce the notions of right F-derivation and left F-derivation of a BCI-algebra and some related properties are explored.

### On eigenvalues and surjectivity using fixed points

JPRM-Vol. 1 (2009), Issue 1, pp. 171 – 175 Open Access Full-Text PDF
Anoop. S. K. , K. T. Ravindran
Abstract: Here first we will prove the existence of fixed points for a weakly continuous, strictly quasi bounded operator on a reflexive Banach space and a completely continuous, strictly quasi bounded operator on any normed linear space. Using these results we can deduce the existence of eigen values and surjectivity of quasi bounded operator in similar situations.

### 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 $$cdn(G) = b$$.