A Study on Edge-Set Graphs of Certain Graphs (original) (raw)
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A Study on the Edge-Set Graphs of Certain Graphs
Let G(V,E)G(V, E)G(V,E) be a simple connected graph, with ∣E∣=epsilon.|E| = \epsilon.∣E∣=epsilon. In this paper, we define an edge-set graph mathcalGG\mathcal G_GmathcalGG constructed from the graph GGG such that any vertex vs,iv_{s,i}vs,i of mathcalGG\mathcal G_GmathcalGG corresponds to the iii-th sss-element subset of E(G)E(G)E(G) and any two vertices vs,i,vk,mv_{s,i}, v_{k,m}vs,i,vk,m of mathcalGG\mathcal G_GmathcalGG are adjacent if and only if there is at least one edge in the edge-subset corresponding to vs,iv_{s,i}vs,i which is adjacent to at least one edge in the edge-subset corresponding to vk,mv_{k,m}vk,m where s,ks,ks,k are positive integers. It can be noted that the edge-set graph mathcalGG\mathcal G_GmathcalGG of a graph GGG id dependent on both the structure of GGG as well as the number of edges epsilon.\epsilon.epsilon. We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.
The (△,□)-Edge Graph G△,□ of a Graph G
2020
To a simple graph G=(V,E)G=(V,E)G=(V,E), we correspond a simple graph Gtriangle,squareG_{triangle,square}Gtriangle,square whose vertex set is x,y:x,yinV{{x,y}: x,yin V}x,y:x,yinV and two vertices x,y,z,winGtriangle,square{x,y},{z,w}in G_{triangle,square}x,y,z,winGtriangle,square are adjacent if and only if x,z,x,w,y,z,y,winVcupE{x,z},{x,w},{y,z},{y,w}in Vcup Ex,z,x,w,y,z,y,winVcupE. The graph Gtriangle,squareG_{triangle,square}Gtriangle,square is called the (triangle,square)(triangle,square)(triangle,square)-edge graph of the graph GGG. In this paper, our ultimate goal is to provide a link between the connectedness of GGG and Gtriangle,squareG_{triangle,square}Gtriangle,square.
Results on the edge-connectivity of graphs
Discrete Mathematics, 1974
It is shown that if G is a graph of order p 2 2 such that deg u + deg u 2 p-1 for all pairs u, u of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg u _> p for all pairs u, u of nonadjacent vertices, then either p is even and G is isomorphic to Kpj2 X K, or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.
Generalized Vertex Induced Connected Subsets Of A Graph
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In this article, we consider finite, undirected, simple and connected graphs G = (V,E) with vertex set V and edge set E. As such p =| V | and q =| E | denote the number of vertices and edges of a graph G, respectively.
The field of mathematics plays vital role in various fields. One of the important areas in mathematics is graph theory which is used in structural models. This structural arrangements of various objects or technologies lead to new inventions and modifications in the existing environment for enhancement in those fields. This Paper describes the description of graph theory.
On Edge-Distance and Edge-Eccentric graph of a graph
2014
An elementary circuit (or tie) is a subgraph of a graph and the set of edges in this subgraphis called an elementary tieset. The distance d(ei, ej ) between two edges in an undirected graph is defined as the minimum number of edges in a tieset containing ei and ej . The eccentricity eτ (ei) of an edge ei is eτ (ei) = maxej∈Ed(ei, ej ). In this paper, we have introduced the edge - self centered and edge - eccentric graph of a graph and have obtained results on these concepts.
Bull. ICA, 2020
If X is any nonempty set on n ≥ 2 elements we define the set graph Gn to be the graph whose vertices are the 2 − 2 proper subsets of X with two vertices adjacent if and only if their underlying sets are disjoint. We discuss some properties of Gn. In particular we find its clique partition number and its product dimension. We also give bounds for its representation number. We use standard graph theory terminology as given in [13]. A family of subsets S1, S2, . . . of a set S gives a graph in a natural way if we use these sets as vertices and let SiSj for i 6= j be an edge if and only if the corresponding subsets have a nonempty intersection. In [12], Marczewski has established the converse, i.e. for any graph G there is a set S, such that a family of its subsets defines G according to the above description. Erdős, Goodman and Posa in [1] have remarked that one may replace the idea of a nonempty intersection with disjointness of the subsets since the same would then imply Marczewski’s...
Discrete Mathematics, 2003
The distance d(u; v) between two vertices u and v in a nontrivial connected graph G is the length of a shortest u-v path in G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. A vertex v of G is a peripheral vertex if e(v) is the diameter of G. The subgraph of G induced by its peripheral vertices is the periphery Per
On the edge-balance index sets of (p, p+ 1)-graphs
2012
Let G be a simple graph with vertex set V (G) and edge set E(G), and let Z 2 = {0, 1}. Any edge labeling f induces a partial vertex labeling f + : V (G) → Z 2 assigning 0 or 1 to f + (v), v being an element of V (G), depending on whether there are more 0-edges or 1edges incident with v, and no label is given to f + (v) otherwise. For each i ∈ Z 2 , let v f (i) = |{v ∈ V (G) : f + (v) = i}| and let e f (i) = |{e ∈ E(G) : f (e) = i}|. An edge-labeling f of G is said to be edge friendly if {|e f (0) − e f (1)| ≤ 1. The edge-balance index set of the graph G is defined as EBI(G) = {|v f (0) − v f (1)| : f is edge-friendly.}. In this paper, we investigate and present results concerning the edge-balance index sets of (p, p + 1)-graphs.