Computing Gutman Connection Index of Thorn Graphs

Chemical structural formula can be represented by chemical graphs in which atoms are considered as vertices and bonds between them are considered as edges. A topological index is a real value that is numerically obtained from a chemical graph to predict its various physical and chemical properties. Thorn graphs are obtained by attaching pendant vertices to the different vertices of a graph under certain conditions. In this paper, a numerical relation between the Gutman connection (GC) index of a graph and its thorn graph is established. Moreover, the obtained result is also illustrated by computing the GC index for the particular families of the thorn graphs such as thorn paths, thorn rods, thorn stars, and thorn rings.

Domination integrity of some graph classes

The stability of a communication network has a great importance in network design. There are several vulnerability measures used to determine the resistance of network to the disruption in this sense. Domination theory provides a model to measure the vulnerability of a graph network. A new vulnerability measure of domination integrity was introduced by Sundareswaran in his Ph.D. thesis (Parameters of vulnerability in graphs (2010)) and defined as DI(G) = min{|S| + m(G − S):S ∈ V(G)} where m(G − S) denotes the order of a largest component of graph G − S and S is a dominating set of G. The domination integrity of an undirected connected graph is such a measure that works on the whole graph and also the remaining components of graph after any break down. Here we determine the domination integrity of wheel graph W1,n, Ladder graph Ln, Sm,n, Friendship graph Fn, Thorn graph of Pn and Cn which are commonly used graph models in network design.

DIMENSI PARTISI GRAF THORN DARI GRAF RODA W3 DAN W4

Let 𝐺 = (𝑉, 𝐸) be a connected graph and 𝑆 ⊆ 𝑉(𝐺). For a vertex v ∈ V(G) and an ordered k-partition Π = {𝑆1 , 𝑆2 , … , 𝑆𝑘 } of 𝑉(𝐺), the representation of v with respect to Π is the k-vector 𝑟(𝑣|𝛱 = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), . . . , 𝑑(𝑣, 𝑆𝑘)), where d(v,Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if for every two vertices 𝑢, 𝑣  𝑉(𝐺), the representation 𝑟(𝑢|П)  𝑟(𝑣|Π). The minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). The wheel graph 𝑊𝑛 𝑜𝑛 𝑛 + 1 vertices with 𝑉(𝑊𝑛) = {𝑣0, 𝑣1, . . . , 𝑣𝑛}. Let 𝑙2 ,𝑙2 ,… ,𝑙𝑛be non-negative integers, 𝑙𝑖 ≥ 1, for 𝑖  {0,1,2, . . . , 𝑛}. The thorn graph of the graph Wn, with parameters 𝑙0 ,𝑙1 ,… ,𝑙𝑛 is obtained by attaching li new vertices of degree one to the vertex vi of the graph Wn. The thorn graph is denoted by 𝑇ℎ(𝑊𝑛,𝑙0 ,𝑙1 ,… ,𝑙𝑛). In this paper we give the upper bounds for the partition dimension of 𝑊3 and 𝑊4 denoted by 𝑝𝑑(𝑇ℎ(𝑊3 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 )) and 𝑝𝑑(𝑇ℎ(𝑊4 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 ,𝑙4 )). Keywords : Partition Dimension, Resolving Partition, Thorn Graph, Wheel Graph.

Exponential Independence Number of Some Graphs

Let [Formula: see text] be a graph and [Formula: see text]. We define by [Formula: see text] the subgraph of [Formula: see text] induced by [Formula: see text]. For each vertex [Formula: see text] and for each vertex [Formula: see text], [Formula: see text] is the length of the shortest path in [Formula: see text] between [Formula: see text] and [Formula: see text] if such a path exists, and [Formula: see text] otherwise. For a vertex [Formula: see text], let [Formula: see text] where [Formula: see text]. Jäger and Rautenbach [27] define a set [Formula: see text] of vertices to be exponential independent if [Formula: see text] for every vertex [Formula: see text] in [Formula: see text]. The exponential independence number [Formula: see text] of [Formula: see text] is the maximum order of an exponential independent set. In this paper, we give a general theorem and we examine exponential independence number of some tree graphs and thorn graph of some graphs.

Hyper Zagreb indices of composite graphs

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Similarly, the hyper Zagreb index is defined as the sum of square of degree of vertices over all the edges. In this paper, First we obtain the hyper Zagreb indices of some derived graphs and the generalized transformations graphs. Finally, the hyper Zagreb indices of double, extended double, thorn graph, subdivision vertex corona of graphs, Splice and link graphs are obtained.

Modified Eccentric Connectivity of Generalized Thorn Graphs

The thorn graph GT of a given graph G is obtained by attaching t(>0) pendent vertices to each vertex of G. The pendent edges, called thorns of GT, can be treated as P2 or K2, so that a thorn graph is generalized by replacing P2 by Pm and K2 by Kp and the respective generalizations are denoted by GPm and GKp. The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph GT and the generalized thorn graphs GKp and GPm.