Square graph in graph theory

Definition 26 Let G G1 G2. Families of graphs derived from classical geometries over finite fields.

As used in graph theory the term graph does not refer to data charts such as line graphs or bar graphs.

Square graph in graph theory. The square of an oriented graph is a graph G whose vertex set V G is the same as the vertex set V G of G. An ordered pair of vertices u w is in the arc set A G of G if and only if there exists a vertex v in G and consequently in G such that u v and v w are arcs in G. This undirected graph called the square graph is defined in the following equivalent ways.

It is the cycle graph on 4 vertices denoted. It is the complete bipartite graph It is the 2-dimensional hypercube graph. For a given graph G the square graph G2is a graph on the same vertex set but in which two vertices are adjacent if and only if they are at distance at most 2 in G.

Definition 23 Square sum labeling. The graph G is said to be a square sum graph strongly square sum graph if G admits a square sum strongly square sum labeling. In this paper we initiate a study of graphs which are square sum.

A graph G is defined as G V E Where V is a set of all vertices and E is a set of all edges in the graph. Example 1 In the above example ab ac cd and bd are the edges of the graph. One reason graph theory is such a rich area of study is that it deals with such a fundamental concept.

Any pair of objects can either be related or not related. What the objects are and what related means varies on context and this leads to many applications of graph theory. A graph which admits square difference labeling is called square difference graph.

In this paper I discussed the square difference labeling is admitted for some graphs like cycles complete graphs cycle cactus ladder lattice grids wheels quadrilateral snakes the graph G K 2 m K 1. Keywords Any graph which admits square. Penn State Math 485 Lecture Notes Version 15 Christopher Gri n.

Are represented by points or squares circles triangles etc and edges are represented by lines connecting vertices4 22 A self-loop is an edge in a graph Gthat contains exactly one vertex. A graphG VE withpvertices andqedges is said to be square sum graph if there exists a bijection mappingfVG 012p1 such that the induced functionfEG N defined byfuv fu2 fv2 for every. A graph with square sum labeling is called square sum graph.

Other terminology and notations in graph theory I follow West 5Some brief summary of definitions which are useful for the present investigations. Definition11 Let G VGEG be a graph is said to be a square sum graph 4 if there exist a bijection f. In a graph no two adjacent vertices adjacent edges or adjacent regions are colored with minimum number of colors.

This number is called the chromatic number and the graph is called a properly colored graph. While graph coloring the constraints that are set on the graph are colors order of coloring the way of assigning color etc. Database of strongly regular graphs.

Database of distance regular graphs. Families of graphs derived from classical geometries over finite fields. Various families of graphs.

1-skeletons of Platonic solids. Sn graphs are square difference graphs. Square difference labeling square difference graph.

Introduction All graphs in this paper are simple finite undirected and nontrivial graph GV E with vertex set V and the edge set E. For graph theoretic terminology we refer to Harary 2. A dynamic survey on graph labeling is regularly updated by Gallian 3.

As used in graph theory the term graph does not refer to data charts such as line graphs or bar graphs. Instead it refers to a set of vertices that is points or nodes and of edges or lines that connect the vertices. When any two vertices are joined by more than one edge the graph is called a multigraph.

A graph which admits square difference labeling is called square difference graph. Definition 26 Let G G1 G2. Gn be graph for n 2.

Then the graph is reduced by adding an edge from Gi to Gi1 for i1 to n-1 is called the path union of G. The square of G written G2 is that graph whose adjacency matrix is As. We shall assume that G has no multiple edges or self-loops since their presence does not alter the square of G.

This paper presents a solution to the problem of characterizing graphs that have at least one square-root graph. We shall need a few definitions. The percentage of nodes directly connected in the entire graph is thus a measure of reachability.

An isolate is a node without connections degree equals to 0. The difference between in-degree and out-degree in a directed graph digraph may underline interesting.