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Keywords = the octagonal grid O n m

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10 pages, 663 KiB  
Article
Total and Double Total Domination on Octagonal Grid
by Antoaneta Klobučar and Ana Klobučar Barišić
Axioms 2024, 13(11), 792; https://doi.org/10.3390/axioms13110792 - 16 Nov 2024
Cited by 1 | Viewed by 1421
Abstract
A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number [...] Read more.
A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number γkt(G) is the cardinality of the smallest k-total dominating set. For k=1,2, the k-total dominating number is called the total and the double total dominating number, respectively. In this paper, we determine the exact values for the total domination number on a linear and on a double octagonal chain and an upper bound for the total domination number on a triple octagonal chain. Furthermore, we determine the exact values for the double total domination number on a linear and on a double octagonal chain and an upper bound for the double total domination number on a triple octagonal chain and on an octagonal grid Om,n,m3,n3. As each vertex in the octagonal system is either of degree two or of degree three, there is no k-total domination for k3. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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14 pages, 341 KiB  
Article
Computing Eccentricity Based Topological Indices of Octagonal Grid O n m
by Xiujun Zhang, Muhammad Kamran Siddiqui, Muhammad Naeem and Abdul Qudair Baig
Mathematics 2018, 6(9), 153; https://doi.org/10.3390/math6090153 - 31 Aug 2018
Cited by 18 | Viewed by 3292
Abstract
Graph theory is successfully applied in developing a relationship between chemical structure and biological activity. The relationship of two graph invariants, the eccentric connectivity index and the eccentric Zagreb index are investigated with regard to anti-inflammatory activity, for a dataset consisting of 76 [...] Read more.
Graph theory is successfully applied in developing a relationship between chemical structure and biological activity. The relationship of two graph invariants, the eccentric connectivity index and the eccentric Zagreb index are investigated with regard to anti-inflammatory activity, for a dataset consisting of 76 pyrazole carboxylic acid hydrazide analogs. The eccentricity ε v of vertex v in a graph G is the distance between v and the vertex furthermost from v in a graph G. The distance between two vertices is the length of a shortest path between those vertices in a graph G. In this paper, we consider the Octagonal Grid O n m . We compute Connective Eccentric index C ξ ( G ) = v V ( G ) d v / ε v , Eccentric Connective Index ξ ( G ) = v V ( G ) d v ε v and eccentric Zagreb index of Octagonal Grid O n m , where d v represents the degree of the vertex v in G. Full article
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