• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.221-229
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• 2018

The first Zagreb eccentricity index $E_1(G)$ of a graph G is defined as the sum of squares of the eccentricities of the vertices. In this paper some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index are computed.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.365-377
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• 2015

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n₁ $\leq$ n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. 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.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.663-680
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• 2024

The zero divisor graph is the most basic way of representing an algebraic structure as a graph. For any commutative ring R, each element is a vertex on the zero divisor graph and two vertices are defined as adjacent if and only if the product of those vertices equals zero. In this research, we determine some topological indices such as the Wiener index, the edge-Wiener index, the hyper-Wiener index, the Harary index, the first Zagreb index, the second Zagreb index, and the Gutman index of zero divisor graph of integers modulo prime power and its direct product.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.327-335
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• 2011

Explicit formulas are given for the first and second Zagreb index, degree-distance and Wiener-type invariants of splice and link of graphs. As a consequence, the first and second Zagreb coindex of these classes of composite graphs are also computed.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.959-977
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• 2022

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.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.401-408
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• 2015

The first and second Zagreb indices of a graph G are defined as $M_1(G)={\sum}_{{\nu}{\in}V}d_G({\nu})^2$ and $M_2(G)={\sum}_{u{\nu}{\in}E(G)}d_G(u)d_G({\nu})$. where $d_G({\nu})$ is the degree of the vertex ${\nu}$. G is called a k-apex tree if k is the smallest integer for which there exists a subset X of V (G) such that ${\mid}X{\mid}$ = k and G-X is a tree. In this paper, we determine the maximum Zagreb indices in the class of all k-apex trees of order n and characterize the corresponding extremal graphs.

• The Pure and Applied Mathematics
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• v.26 no.3
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• pp.177-188
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• 2019

Suppose G is a molecular graph with edge set E(G). The hyper-Zagreb index of G is defined as $HM(G)={\sum}_{uv{\in}E(G)}[deg_G(u)+deg_G(v)]^2$, where $deg_G(u)$ is the degree of a vertex u in G. In this paper, all chemical trees of order $n{\geq}12$ with the first twenty smallest hyper-Zagreb index are characterized.

• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.25-32
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• 2017

Let G be a graph and k is a positive integer. Hammack and Livesay in [The inner power of a graph, Ars Math. Contemp., 3 (2010), no. 2, 193-199] introduced a new graph operation $G^{(k)}$, called the $k^{th}$ inner power of G. In this paper, it is proved that if G is bipartite then $G^{(2)}$ has exactly three components such that one of them is bipartite and two others are isomorphic. As a consequence the edge frustration index of $G^{(2)}$ is computed based on the same values as for the original graph G. We also compute the first and second Zagreb indices and coindices of $G^{(2)}$.

• Journal of applied mathematics & informatics
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• v.31 no.3_4
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• pp.533-544
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• 2013

The multiplicative version of Wiener index (${\pi}$-index), proposed by Gutman et al. in 2000, is equal to the product of the distances between all pairs of vertices of a (molecular) graph G. In this paper, we first present some sharp bounds in terms of the order and other graph parameters including the diameter, degree sequence, Zagreb indices, Zagreb coindices, eccentric connectivity index and Merrifield-Simmons index for ${\pi}$-index of general connected graphs and trees, as well as a Nordhaus-Gaddum-type bound for ${\pi}$-index of connected triangle-free graphs. Then we study the behavior of ${\pi}$-index upon the case when removing a vertex or an edge from the underlying graph. Finally, we investigate the extremal properties of ${\pi}$-index within the set of trees and unicyclic graphs.

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.159-167
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• 2024

The Topological indices are known as Mathematical characterization of molecules. In this paper, we have studied line graph of subdivision and semi-total point graph of triangular benzenoid, polynomino chains of 8-cycles and graphene sheet through forgotten and first Zagreb indices.