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The Zagreb indices of bipartite graphs with more edges
Kexiang Xu,Kechao Tang,Hongshuang Liu,Jinlan Wang 한국전산응용수학회 2015 Journal of applied mathematics & informatics Vol.33 No.3
For a (molecular) graph, the first and second Zagreb indices ($M_1$ and $M_2$) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajsti\'{c}. The first Zagreb index $M_1$ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index $M_2$ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let ${K}_{n_1,n_2}^p$ with $n_1\leq n_2$, $n_1+n_2=n$ and $p<n_1$ be the set of bipartite graphs obtained by deleting $p$ edges from complete bipartite graph $K_{n_1,n_2}$. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from ${K}_{n_1,n_2}^p$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from ${K}_{n_1,n_2}^p$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.
THE ZAGREB INDICES OF BIPARTITE GRAPHS WITH MORE EDGES
XU, KEXIANG,TANG, KECHAO,LIU, HONGSHUANG,WANG, JINLAN The Korean Society for Computational and Applied M 2015 Journal of applied mathematics & informatics Vol.33 No.3
For a (molecular) graph, the first and second Zagreb indices (M<sub>1</sub> and M<sub>2</sub>) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M<sub>1</sub> is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M<sub>2</sub> is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n<sub>1</sub> $\leq$ n<sub>2</sub>, n<sub>1</sub> + n<sub>2</sub> = n and p < n<sub>1</sub> be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K<sub>n1,n2</sub>. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.
UPHILL ZAGREB INDICES OF SOME GRAPH OPERATIONS FOR CERTAIN GRAPHS
SALEH, ANWAR,BAZHEAR, SARA,MUTHANA, NAJAT The Korean Society for Computational and Applied M 2022 Journal of applied mathematics & informatics Vol.40 No.5-6
The topological indices are numerical parameters which determined the biological, physical and chemical properties based on the structure of the chemical compounds. One of the recently topological indices is the uphill Zagreb indices. In this paper, the formulae of some uphill Zagreb indices for a few graph operations of some graphs have been derived. Furthermore, the precise formulae of those indices for the honeycomb network have been found along with their graphical profiles.
Relations between distance-based and degree-based topological indices
Das, K.Ch.,Gutman, I.,Nadjafi-Arani, M.J. Elsevier [etc.] 2015 Applied Mathematics and Computation Vol.270 No.-
Let W, Sz, PI, and WP be, respectively, the Wiener, Szeged, PI, and Wiener polarity indices of a molecular graph G. Let M<SUB>1</SUB> and M<SUB>2</SUB> be the first and second Zagreb indices of G. We obtain relations between these classical distance- and degree-based topological indices.
K^th-eccentricity index of graphs
Veena Mathad,PARVATHI,Ismail Naci CANGUL 장전수학회 2022 Proceedings of the Jangjeon mathematical society Vol.25 No.2
The molecular topological descriptors are the numerical in- variants of a molecular graph and are very useful for predicting their physical properties, chemical reactivity and bioactivity. A variety of such indices are studied and used in theoretical chemistry and phar- maceutical research related to drugs and also in different fields. The main classes of topological graph indices are those based on vertex de- grees, distances, and graph parameters like eccentricity. In this pa- per, we introduce kth-eccentricity index of graphs. Also we compute kth-eccentricity index of some standard graphs including some wind- mill graphs and molecular graphs of cycloalkenes. Further, we obtain lower and upper bounds for the kth-eccentricity index in terms of other topological indices.
A. R. Bindusree,V. Lokesha,P. S. Ranjini 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.3
For a connected graph H of order at least 3, the H-line graph HL(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of HL(G) are adjacent if and only if the corresponding edges are adjacent in G and belong to a common copy of H. Topological indices have a prominent place in chemical graph theory. Topological index is a type of a molecular descriptor that is cacluated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. It does not depend on the labelling or the pictorial representation of a graph. There are several topological indices have been defined and many of them have found application in various fields of science an technology. The first and second Zagreb indices are amongst the oldest and best known topological indices.
Some topological indices of edge-neighborhood corona of two graphs
Sumithra,Malpashree R.,Rakshith B. R. 장전수학회 2017 Advanced Studies in Contemporary Mathematics Vol.27 No.3
In this paper, we introduce edge-neighborhood corona of two graphs and compute its Wiener index, degree distance index and Gutman index.
Inverse problem for the first entire Zagreb index
Muge TOGAN,Aysun YURTTAS,Ismail Naci CANGUL 장전수학회 2019 Advanced Studies in Contemporary Mathematics Vol.29 No.2
The inverse problem for topological graph indices is about the exis- tence of a graph having its index value equal to a given non-negative integer. In this paper, we study the problem for the rst entire Zagreb index. We will rst show that the rst entire Zagreb index must be even for any graph G, and can take all positive even integer values except 4; 6; 10; 12; 14; 18; 20; 22; 26; 28; 30; 36; 38 and 46.
Topological Indices on Model Graph Structure of Alveoli in Human Lungs
V. Lokesha,A. Usha,P. S. Ranjini,K. M. Devendraiah 장전수학회 2015 Proceedings of the Jangjeon mathematical society Vol.18 No.4
Alveoli are hollow cavitites within the human body. In the present study, alveoli of human lungs are considered which are healthy and also when aected by emphysema loss of elasticity resulting in breathing diculties. It is interesting to apply graph theory with a view to test and predict the status of alveoli in both healthy lungs and when aected. The model developed can be used for further advancement in the medical eld for any diagnosis with respect to the lung diseases. The purpose of this paper is to investigation of Alveoli in Human lungs using concept of Topological indices. Here we have attempted to use double graphs by considering the alveoli as a connected graph. Topological indices are determined for healthy and ruptured alveoli by using graph operator called double graphs. PI and Szeged indices have been found for a healthy alveoli as was modelled earlier. Four graph operators namely S(G) , R(G) , Q(G) and L(G) are obtained and their respective PI and Szeged indices are studied. Finally, a comparative study is made for all the operators with respect to the aected alveoli in order to better facilitate the modeling to help detect the defects early.
Operations on Dutch windmill graph of topological indices
V.Lokesha,Sushmitha Jain,T. Deepika,A. Sinan Cevik 장전수학회 2018 Proceedings of the Jangjeon mathematical society Vol.21 No.3
Topological indices are well studied in recent years. These are useful tools in studying Quantitative Structure Activity Relationship (QSAR) and Quantitative Structure Property Relationship (QSPR). The main goal of this paper is to concentrate the investigation on generalized version of Dutch windmill graph of certain graph operators in terms of topological indices, for instance, symmetric division deg index, rst and second Zagreb indices.