Define the average path length in a connected graph as the sum of the length of the shortest path between all pairs of nodes, divided by the total number of pairs of nodes. Letting SN denote the sum of the shortest path lengths between all pairs of nodes in a complete m-ary tree of depth N , we derive a first-order linear but non-homogeneous recurrence relation for SN , from which a closed-form...

Recently three prime cordial labeling behavior of path, cycle, complete graph, wheel, comb, subdivison of a star, bistar, double comb, corona of tree with a vertex, crown, olive tree and other standard graphs were studied. Also four prime cordial labeling behavior of complete graph, book, flower were studied. In this paper, we investigate the four prime cordial labeling behavior of corona of wh...

The Hamiltonian cycle problem is to decide whether a given graph has a Hamiltonian cycle. Bertossi and Bonuccelli (1986, Information Processing Letters, 23, 195200) proved that the Hamiltonian Cycle Problem is NP-Complete even for undirected path graphs and left the Hamiltonian cycle problem open for directed path graphs. Narasimhan (1989, Information Processing Letters, 32, 167-170) proved tha...

In this paper, we analyze the secure connectivity in Shotgun cellular systems (SCS: Wireless communication systems with randomly placed base stations) by Poisson intrinsically secure communication graph (IS-graph), i.e., a random graph which describes the connections that are secure over a network. For a base-station in SCS, a degree of secure connections is determined over two channel models: ...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph G with n edges that yields cyclic G-decompositions of the complete graph K2nt+1 was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a γ-labeling. Here we show that the...

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergr...

The path-partition problem is to find a minimum number of vertex-disjoint paths that cover all vertices of a given graph. This paper studies the path-partition problem from an algorithmic point of view. As the Hamiltonian path problem is NP-complete for many classes of graphs, so is the path-partition problem. The main result of this paper is to present a linear-time algorithm for the path-part...

The slope variety of a graph is an algebraic set whose points correspond to drawings of that graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on n vertices over F2, and the complementreducible graphs on n vertices.

1. INTRODUCTION. A Hamiltonian cycle in a graph is a path in the graph which visits each vertex exactly once and returns to the starting vertex. Let í µí°¾ í µí±› be a weighted complete graph with í µí±› vertices. We define the weight of an edge as the square of the distance between two end points of the edge. The weight of a path í µí±ƒ is the sum of the weights of all edges in the path and de...