Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

The Total Open Monophonic Number of a Graph

For a connected graph G of order n >- 2, a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G).  A  monophonic set of cardinality m(G) is called a m-set of G. A set S of vertices  of a connected graph G is an open monophonic set of G if for each vertex v  in G, either v is an extreme vertex of G and v ˆˆ? S, or v is an internal vertex of a x-y monophonic path for some x, y ˆˆ? S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). A connected open monophonic set of G is an open monophonic set S such that the subgraph < S > induced by S is connected. The minimum cardinality of a connected open monophonic set of G is the connected open monophonic number of G and is denoted by omc(G). A total open monophonic set of a graph G is an open monophonic set S such that the subgraph < S > induced by S contains no isolated vertices. The minimum cardinality of a total open monophonic set of G is the total open monophonic number of G and is denoted by omt(G). A total open monophonic set of cardinality omt(G) is called a omt-set of G. The total open monophonic  numbers of certain standard graphs are determined. Graphs with total open monphonic number 2 are characterized. It is proved that if G is a connected graph such that omt(G) = 3 (or omc(G) = 3), then G = K3 or G contains exactly two extreme vertices. It is proved that for any integer n  3, there exists a connected graph G of order n such that om(G) = 2, omt(G) = omc(G) = 3. It is proved that for positive integers r, d and k  4 with 2r, there exists a connected graph of radius r, diameter d and total open monophonic number k. It is proved that for positive integers a, b, n with 4 <_ a<_ b <_n, there exists  a connected graph G of order n such that omt(G) = a and omc(G) = b

The connected detour monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path, for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A connected detour monophonic set of G is a detour monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected detour monophonic set of G is the connected detour monophonic number of G and is denoted by dmc(G). We determine bounds for dmc(G) and characterize graphs which realize these bounds. It is shown that for positive integers r, d and k ≥ 6 with r < d, there exists a connected graph G with monophonic radius r, monophonic diameter d and dmc(G) = k. For each triple a, b, p of integers with 3 ≤ a ≤ b ≤ p − 2, there is a connected graph G of order p, dm(G) = a and dmc(G) = b. Also, for every pair a, b of positive integers with 3 ≤ a ≤ b, there is a connected graph G with mc(G) = a and dmc(G) = b, where mc(G) is the connected monophonic number of G.The first author is partially supported by DST Project No. SR/S4/MS:570/09.Publisher's Versio

The restrained monophonic number of a graph

A set S of vertices of a connected graph G is a monophonic set of G if each vertex v of G lies on a x−y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). We determine bounds for it and determine the same for some special classes of graphs. Further, several interesting results and realization theorems are proved.Publisher's Versio

A stochastic model for sero conversion times of HIV transmission

This paper focuses on the study of a Stochastic Model for predicting the seroconversion time of HIV transmission. As the immune capacities of an individual vary and also have its own resistance, the antigenic diversity threshold is different for different person. We propose a stochastic model to study the damage process acting on the immune system that is non- linear. The mean of seroconversion time of HIV and its variance are derived. A numerical example is given to illustrate the seroconversion times of HIV transmission

Minimal restrained monophonic sets in graphs

For a connected graph G = (V, E) of order at least two, a restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V −S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). A restrained monophonic set S of G is called a minimal restrained monophonic set if no proper subset of S is a restrained monophonic set of G. The upper restrained monophonic number of G, denoted by m+r (G), is defined as the maximum cardinality of a minimal restrained monophonic set of G. We determine bounds for it and find the upper restrained monophonic number of certain classes of graphs. It is shown that for any two positive integers a, b with 2 ≤ a ≤ b, there is a connected graph G with mr(G) = a and m+r (G) = b. Also, for any three positive integers a, b and n with 2 ≤ a ≤ n ≤ b, there is a connected graph G with mr(G) = a, m+r (G) = b and a minimal restrained monophonic set of cardinality n. If p, d and k are positive integers such that 2 ≤ d ≤ p − 2, k ≥ 3, k 6= p − 1 and p − d − k ≥ 0, then there exists a connected graph G of order p, monophonic diameter d and m+r (G) = k.The third author’s research work has been supported by NBHM, India.Publisher's Versio

Restrained Double Monophonic Number of a Graph

For a connected graph G of order at least two, a double monophonic set S of a graph G is a restrained double monophonic set if either S=V or the subgraph induced by V−S has no isolated vertices. The minimum cardinality of a restrained double monophonic set of G is the restrained double monophonic number of G and is denoted by dmr(G). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers a,b,c with 3≤a≤b≤c, there is a connected graph G with m(G)=a, mr(G)=b and dmr(G)=c, where m(G) is the monophonic number and mr(G) is the restrained monophonic number of a graph G.The second author research work was supported by National Board for Higher Mathematics, INDIA (Project No. NBHM/R.P.29/2015/Fresh/157).The authors are thankful to the reviewers for their useful comments for the improvement of this paper

Pulsed microwave assisted hot air drying of nutmeg mace for better colour retention

A study was conducted on application of novel drying technology for better color retention of mace. Pulsed microwave assisted hot air drying was investigated at three different power levels 0.5 kW, 1 kW and 1.445 kW with 30 seconds pulsation at a hot air temperature of 45°C and the color values of mace were compared with the market and fresh sample using colorimeter. Further, the major flavor compound, myristicin in mace was analysed. &nbsp