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Rainbow Vertex Connection Number of a Class of Triangular Snake Graph
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
Dharamvirsinh Parmar, Bharat Suthar
The rainbow connection number was introduced by Chartrand, Johns, McKeon, and Zhang in [4]. It has applications in transferring information of high security in multicomputer networks. We refer the reader to [7; 9] for details. The rainbow vertex connection number of a graph, denoted by rvc(G) is the smallest number of colors that are needed in order to make graph G rainbow vertex connected. M. Krivelevich and R. Yuster in [7] gave the lower bound for rvc(G), diam (G) − 1 ≤ rvc(G). Li. Hengzhe and Ma. Yingbin [6] discussed rainbow connection number and graph operation. Also Annammal and Mercy [2] derive the rainbow connection number of shadow graphs. Dian N. S. Simamora and A. N. M.Salman [3] discussed the rainbow (vertex) coloring of a pencil graph. In this chapter, we focus on the rainbow vertex connection number of the triangular snake graph, double triangular snake graph, triple triangular snake graph, alternating triangular snake, double alternating triangular snake and quadrilateral graph.
Structure and Representation
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
In Exercises 2.1.1 through 2.1.6, find all possible isomorphism types of the given kind of simple graph. 2.1.1S A 4-vertex tree.2.1.2 A 4-vertex connected graph.2.1.3 A 5-vertex tree.2.1.4 A 6-vertex tree.2.1.5 A 5-vertex graph with exactly three edges.2.1.6 A 6-vertex graph with exactly four edges.
Atom-Bond Connectivity Index
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Chen and Guo (2011) characterized the catacondensed hexagonal systems with the maximum and minimum ABC indices. They also proved that the ABC index of a graph decreases when any edge is deleted. This result implies that Chen & Guo (2011), among all n-vertex graphs, the complete graph Kn has the maximal ABC value and among all n-vertex connected graphs, the graph with the minimal ABC index is a tree.
Aggregation of clans to speed-up solving linear systems on parallel architectures
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
Dmitry A. Zaitsev, Tatiana R. Shmeleva, Piotr Luszczek
Aggregation of subset of clans replaces a sub-graph induced by a subset of vertices by a new single vertex according to the following rules: and for each vertex connected with a vertex in A, all edges are replaced by an edge of the summary weight the aggregated clan weights are calculated as follows: and
Modelling epidemics on d-cliqued graphs
Published in Letters in Biomathematics, 2018
Laura P. Schaposnik, Anlin Zhang
A rooted tree is a tree in which one vertex has been designated the root. In such trees, the parent of a vertex v is the vertex connected to it on the path to the root , and v is called a child of the parent vertex. The height is the length, in number of edges, from a terminal vertex to the root. We assume that all terminal vertices of the d-regular tree have the same height and call this the height of the d-regular tree. The network diameterL of a d-regular tree is the number of edges in the longest path between two vertices. Regular trees are closely related to d-ary trees (Figure 2).
Path histogram distance and complete subtree histogram distance for rooted labelled caterpillars
Published in Journal of Information and Telecommunication, 2020
Taiga Kawaguchi, Takuya Yoshino, Kouichi Hirata
Let be a star, that is, consists of a single vertex, and the caterpillar obtained by adding new vertex connected to the root of . Then, since , it holds that . On the other hand, it is obvious that , so the statement 1 holds. Furthermore, the same caterpillars and in the proof of Theorem 3.7.2 also satisfy the statement 2.