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Geometric-Arithmetic Index
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
In graph theory, a regular graph is a graph in which every vertex has the same degree. A graph which is not regular is said to be irregular. In order to establish “how irregular” a given graph is, several irregularity measures, denoted here by irr(G), have been proposed. Each of such measures must satisfy the condition irr(G) ≥ 0 and irr(G) = 0 if the graph G is regular. In chemical applications, quantifying the irregularity of molecular graphs and of biomolecular networks seems to be of marginal importance. Gutman et al. (2014) proposed an irregularity measure by means of the GA index. Since, for a connected graph G with m edges, GA(G) ≤ m with equality if and only if G is regular (Vukičević & Furtula, 2009), so irrGA(G)=1−GA(G)m.
Introduction to Graph Theory
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
A graph is a regular graph if the degree of every vertex is the same integer. A k-regular graph is one for which the degree of each vertex is k. A 3-regular graph is often called a cubic graph. The complete graph Kn $ K_n $ is an (n-1) $ (n-1) $ -regular graph.
Faulty diagnosability and g-extra connectivity of DQcube
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
Let G be an n-regular -graph and is a 2-path or a 3-cycle in . If G satisfies the following conditions For any with G−F has a large component and small components which contain at most two vertices in total; if G contains no 5-cycle, and otherwise; then .
Tensor and Cartesian products for nanotori, nanotubes and zig–zag polyhex nanotubes and their applications to 13C NMR spectroscopy
Published in Molecular Physics, 2021
Medha Itagi Huilgol, V. Sriram, Krishnan Balasubramanian
Let be a distance degree regular graph. We are given that G and H are two bipartite graphs. Suppose G is not a DDR graph, then there exist at least two vertices u and v having different distance degree sequences, and , with , in G. Let k, the minimum value of i, such that Let w be any vertex of H, having the distance degree sequence and be the number of vertices at distance k from w in H.
A survey on graphs with convex quadratic stability number
Published in Optimization, 2020
Now it is worth mention the concept of -regular set, introduced in [32], which is a vertex subset S of a graph G, inducing a κ-regular subgraph such that every vertex out of S has τ neighbours in S, that is, for any vertex we have For convenience, when G is a p-regular graph, the whole vertex set is considered a -regular set. For instance, considering the Petersen graph depicted in Figure 3, the following -regular sets are obtained. The set is -regular.The set is -regular.The set is -regular.