A graph G is said to be Smarandachely harmonic graph with property P if its vertices can be labeled 1, 2, · · · , n such that the function fP : A → Q defined by fp(H) = ∏ v∈V (H) f(v) ∑ v∈V (H) f(v) , H ∈ A is injective. Particularly, if A is the collection of all paths of length 1 in G (That is, A = E(G)), then a Smarandachely harmonic graph is called Strongly harmonic graph. In this paper we ...

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced s...

We investigate paths, cycles and wheels in graphs with independence number of at most 2, in particular we prove theorems characterizing ali such graphs which are hamiltonian. Ramsey numbers of the form R (G, K 3)' for G being a path, a cycle or a wheel, are known to be 2n (G) 1, except for some small cases. In this paper we derive and count all critical graphs for these Ramsey numbers. 1. Notat...

The detergents are household chemical cleaning compound used in wide range of our daily life for diverse purposes. Wheel is a common detergent produced and largely Bangladesh. Some the constituents show high biological activity has tendency to accumulate organisms, making adverse effects possible even at very low level exposure. freshwater fish like Channa punctatus (Bloch, 1793) highly sensiti...

The Ramsey number for a graph G versus a graph H, denoted by R(G,H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of stars versus small cycle and small wheel. We show that if ni ≥ 3 for i = 1, 2, . . . , k and ni ≥ ni+1 ≥ √ ni − 2, then R( ∪k i=...

Let G be a graph and p ∈ [1,∞]. The parameter fp(G) is the least integer k such that for all m and all vectors (rv)v∈V (G) ⊆ R , there exist vectors (qv)v∈V (G) ⊆ R k satisfying ‖rv − rw‖p = ‖qv − qw‖p, for all vw ∈ E(G). It is easy to check that fp(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [9], there are a finite number of excluded m...

A bar 1-visibility drawing of a graph G is a drawing of G where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an en...

The Ramsey number for a graph G versus a graph H, denoted by R(G, H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of trees versus small cycle and small wheel. We show that if ni is odd and 2ni+1 ≥ ni for every i, then R( Sk i=1 Tni , W4) = R(Tn...