• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.595-608
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• 2013

Let G = (V, E) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V (G) and E = E(G), respectively. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The distance between the vertices $u$ and $v$ in V (G) of graph G is the number of edges in a shortest path connecting them, we denote by $d(u,v)$. In graph theory, we have many invariant polynomials for a graph G. In this paper, we focus on the Schultz polynomial, Modified Schultz polynomial, Hosoya polynomial and their topological indices of a molecular graph circumcoronene series of benzenoid $H_k$ and specially third member from this family. $H_3$ is a basic member from the circumcoronene series of benzenoid and its conclusions are base calculations for the Schultz polynomial and Hosoya polynomial of the circumcoronene series of benzenoid $H_k$ ($k{\geq}3$).

• Journal of Applied and Pure Mathematics
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• v.4 no.5_6
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• pp.249-262
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• 2022

Nowadays, it is an important task to find extremal values on any molecular descriptor with respect to different graph parameters. The Sombor index is a novel topological molecular descriptor introduced by Gutman in 2021. The research on determining extremal values for the Sombor index of a graph is very popular recently. In this paper, we present the maximum Sombor index of bicyclic graphs with given matching number. Furthermore, we identify the corresponding extremal bicyclic graphs.

• The Pure and Applied Mathematics
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• v.27 no.4
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• pp.171-186
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• 2020

Long back in 1972, it was shown that the sum of the squares of vertex degrees and the sum of cubes of vertex degrees of a molecular graph both have large correlations with total 𝜋-electron energy of the molecule. Later on, the sum of squares of vertex degrees was named as first Zagreb index and became one of the most studied molecular graph parameter in the field of chemical graph theory. Whereas, the other sum remained almost unnoticed until recently except for a few occasions. Thus it got the name "forgotten" index or F-index. This paper investigates extremal graphs with respect to F-index among the class of bicyclic graphs with n vertices and k pendant vertices, 0 ≤ k ≤ n - 4. As consequences, we obtain the bicyclic graphs with largest and smallest F-indices.

• Proceedings of the Korea Information Processing Society Conference
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• 2022.11a
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• pp.380-382
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• 2022

Extracting informative representation of molecules using graph neural networks(GNNs) is crucial in AI-driven drug discovery. Recently, the graph research community has been trying to replicate the success of self supervised in natural language processing, with several successes claimed. However, we find the benefit brought by self-supervised learning on applying varitional auto-encoders can be potentially effective on molecular data.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.319-330
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• 2016

The hyper-Zagreb index HM(G) of a simple graph G is defined as the sum of the terms (d_u+d_v)² over all edges uv of G, where d_u denotes the degree of the vertex u of G. In this paper, we present several upper and lower bounds on the hyper-Zagreb index in terms of some molecular structural parameters and relate this index to various well-known molecular descriptors.

• 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.

• Molecular & Cellular Toxicology
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• v.1 no.4
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• pp.281-283
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• 2005

Extraction of biologically meaningful data and their validation are very important for toxicogenomics study because it deals with huge amount of heterogeneous data. BINGO is an annotation mining tool for biological interpretation of gene groups. Several statistical modeling approaches using Gene Ontology (GO) have been employed in many programs for that purpose. The statistical methodologies are useful in investigating the most significant GO attributes in a gene group, but the coherence of the resultant GO attributes over the entire group is rarely assessed. BINGO complements the statistical methods with graph-theoretic measures using the GO directed acyclic graph (DAG) structure. In addition, BINGO visualizes the consistency of a gene group more intuitively with a group-based GO subgraph. The input group can be any interesting list of genes or gene products regardless of its generation process if the group is built under a functional congruency hypothesis such as gene clusters from DNA microarray analysis.

• 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 the Korea Convergence Society
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• v.3 no.4
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• pp.15-21
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• 2012

Image segmentation for object to extract data from ultrasound acquired is an essential preprocessing step for the effective diagnosis. Various image segmentation methods have been studied. In this study, interactive image segmentation method by graph cut algorithm is proposed to develop a variety of applications of vascular ultrasound imaging and diagnostics. General imaging and vascular ultrasound imaging segmentation by entering constrain condition such as foreground and background. In the future it will be able to develop new ultrasound diagnostics.

• 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.