Index Of Graph Research Articles

Bounds on The Reduced Sombor Index of Graphs

Bounds on The Reduced Sombor Index of Graphs

Data-driven approaches to study the spectral properties of chemical structures

The molecular energy, which is the sum of all eigenvalues, is crucial in determining the total π-electron energy of conjugated hydrocarbon molecules. We used machine learning techniques to calculate the energy, inertia, nullity, signature, and Estrada index of molecular graphs for bismuth tri-iodide and benzene rings embedded in P-type surfaces within 2D networks. We applied MATLAB to extract the actual eigenvalues from the data and developed general equations for these molecular properties. We then used these equations to estimate the values and compared them to the actual values through graphical analysis. Our results demonstrate the potential of data-driven techniques in predicting molecular properties and enhancing our understanding of spectral theory.

Enumeration of the Hosoya index of pericondensed benzenoid system

In chemical graph theory, topological indices play an important role and have various uses in quantitative structure-property relationships (QSPR) as well as quantitative structure-activity relationships (QSARs). The Hosoya index is one of them, performing a molecular descriptor in mathematical chemistry. Therefore, progressing with investigations requires computing the Hosoya index of distinct molecular graphs. This research presents a computational method for determining the Hosoya index of pericondensed benzenoid systems using the transfer matrix approach and the Hosoya vector.

Sharp lower bounds for the Laplacian Estrada index of graphs

Let G be a simple graph on n vertices, and let λ 1 , λ 2 , … , λ n be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE ( G ) = ∑ i = 1 n e λ i . Consider a graph G with n ≥ 3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue λ n . Let K n , S n , and K p , q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE ( G ) ≥ n e 2 m n + c + e λ n − ( c + 1 ) e λ n c + 1 . Furthermore, we show that the equality holds if and only if G ≅ K ¯ n (the complement of K n ), G ≅ ∪ i = 1 c − 1 K 1 ∪ S c + 1 if n = 2c, or G ≅ K n 2 , n 2 if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175–180.] is not complete. Consequently, we provide a complete version of this lower bound.

Maximal elliptic Sombor index of bicyclic graphs

Maximal elliptic Sombor index of bicyclic graphs

On Topological Indices of Fuzzy Random Graph

On Topological Indices of Fuzzy Random Graph

An Interactive Multi-Modal Query Answering System with Retrieval-Augmented Large Language Models

Retrieval-augmented Large Language Models (LLMs) have reshaped traditional query-answering systems, offering unparalleled user experiences. However, existing retrieval techniques often struggle to handle multi-modal query contexts. In this paper, we present an interactive M ulti-modal Q uery A nswering (MQA) system, empowered by our newly developed multi-modal retrieval framework and navigation graph index, integrated with cutting-edge LLMs. It comprises five core components: Data Preprocessing, Vector Representation, Index Construction, Query Execution, and Answer Generation, all orchestrated by a dedicated coordinator to ensure smooth data flow from input to answer generation. One notable aspect of MQA is its utilization of contrastive learning to assess the significance of different modalities, facilitating precise measurement of multimodal information similarity. Furthermore, the system achieves efficient retrieval through our advanced navigation graph index, refined using computational pruning techniques. Another highlight of our system is its pluggable processing framework, allowing seamless integration of embedding models, graph indexes, and LLMs. This flexibility provides users diverse options for gaining insights from their multi-modal knowledge base. A preliminary video introduction of MQA is available at https://youtu.be/xvUuo2ZIqWk.

Planar Stick Indices of Some Knotted Graphs

Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.

On Sombor Index of Unicyclic and Bicyclic Graphs

Gutman proposed a topological index called the Sombor index, which was defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] in graph [Formula: see text]. In this paper, we determine the second-minimum and second-maximum values of the Sombor index over all the unicyclic graphs of order [Formula: see text] and bicyclic graphs of order [Formula: see text].

Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs

Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs

Extremal graphs with respect to modified Sombor index

The modified Sombor (mSO) index of a given graph [Formula: see text] is symbolized with mSO and define by sum of the weights [Formula: see text], where [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]. In this work, we present the first and second maximum values of the mSO index among all [Formula: see text]-vertex [Formula: see text]-cyclic graphs. We also provide that the mSO index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the mSO index of connected chemical graphs.

Characterizing Finite Groups through Equitable Graphs: A Graph-Theoretic Approach

This paper introduces equitable graphs of Type I associated with finite groups. We investigate the connectedness and some graph-theoretic properties of these graphs for various groups. Furthermore, we establish the novel concepts of the equitable square-free number and the equitable group. Our study includes an analysis of the equitable graphs for specific equitable groups. Additionally, we determine the first, second and forgotten Zagreb topological indices for the equitable graphs of Type I constructed from certain groups. Finally, we derive the adjacency matrix for this graph type built from cyclic p-groups.

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

On the Laplacian index of tadpole graphs

Abstract In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle C k {C}_{k} and a path P n {P}_{n} . Using two different approaches, we show that their Laplacian index converges to Δ 2 Δ − 1 = 9 2 \frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2} as n , k → ∞ n,k\to \infty , where Δ = 3 \Delta =3 is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic, Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory, J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.

Sharp Upper Bound for Harmonic Index of Trees with Given Total Domination Number

Let \( G=(V,E) \) be a simple connected graph with vertex set \( G \) and edge set \( E \). The harmonic index of graph \( G \) is the value \( H(G)=\sum_{uv\in E(G)} \frac{2}{d_u+d_v} \), where \( d_x \) refers to the degree of \( x \). We obtain an upper bound for the harmonic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound.

Similarity search on social networks with incremental graph indexing based on probabilistic inference

PurposeThis paper aims to propose an incremental graph indexing method based on probabilistic inferences in Bayesian network (BN) for approximate nearest neighbor search (ANNS) that adds unindexed queries into the graph index incrementally.Design/methodology/approachThis paper first uses the attention mechanism based graph convolutional network to embed a social network into the low-dimensional vector space, which could improve the efficiency of graph index construction. To add the unindexed queries into the graph index incrementally, this study proposes to learn the rule-based BN from social interactions. Thus, the dependency relations of unindexed queries and their neighbors are represented, and the probabilistic inferences in BN are then performed.FindingsExperimental results demonstrate that the proposed method improves the search precision by at least 5% and search efficiency by 10% compared to the state-of-the-art methods.Originality/valueThis paper proposes a novel method to construct the incremental graph index based on probabilistic inferences in BN, such that both indexed and unindexed queries in ANNS could be addressed efficiently.

On the QSPR Models of some Anti-Malaria Drugs using some Reversed Degree-Based Topological Indices

A topological index is a real number obtained from the chemical structure of a molecular graph that can be used to predict the physiochemical and biological properties of chemical compounds like drugs and benzene structures. This study aimed at proposing a quantitative structure-property relationship (QSPR) model for some physiological and biological properties of some anti-malaria drugs. Some reversed-degree-based topological indices of the molecular graphs of these drugs were computed; hence, a linear regression model analysis of these indices with the empirical values of the said properties of these drugs was carried out and presented. It was found that the topological indices computed for the said drugs have a good correlation with these physicochemical properties.

Innovative graph-based video processing methodology for collapse early warning of historic masonry buildings

In the present study an innovative video processing methodology is proposed for application to vibration data of historic masonry structures. The proposed methodology is based on graph theory and topology analysis applied to magnified videos in search of effective parameters for early-warning signals before the collapse of structures in case of earthquakes. The proposed method was validated through seismic tests of a brick-masonry mockup representing a vault of the mosque in the Palace of the Dey, Algiers. In particular, the seismic tests were carried out at increasing earthquake intensity up to the final collapse of the mockup. After processing the videos of the seismic tests by motion magnification method, the magnified video frames were transformed into a graph of the structure. Finally, several graph indices were calculated and monitored during the vibration. The monitored parameters were analyzed in search of potential threshold values suitable to generate an early warning signal. In particular, the inverse algebraic connectivity provided an early warning signal in the order of a few seconds before collapse. This was validated by comparison with an analogous signal provided at similar time by an accurate displacement lab measurements system based on optical markers positioned at several points of the tested mock-up.

First and second K Banhatti indices of a fuzzy graph applied to decision making

Topological index of a graph is a numerical representation linked to the graph or the molecular structure associated with the graph, shedding light on its topological and structural properties. Different topological indices showcase unique attributes rooted in their degree, spectrum, and distance characteristics. Among these, the first and second K Banhatti indices, introduced by Kulli, involving both vertex and edge degrees, have garnered significant scholarly interest, prompting extensive discussions in graph theory. Within this research paper, a thorough exploration delves into the attributes surrounding the fuzzified first and second K Banhatti indices. This exploration is directed towards several bounds of the index and index of various structures like cycles, complete fuzzy graphs, complete bipartite graphs and stars. Relation between a fuzzy graph and its partial subgraphs, fuzzy graph and its complement graph in connection with the indices is also found. Additionally, the research paper investigates the first and second K Banhatti indices concerning graph operations such as the corona product and coalescence. Additionally, the paper outlines an algorithm designed to accurately compute the first and second K Banhatti indices and proposes an application in decision-making.

On the topological indices and graph operation

As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In [A. Miličević, A. Nikolić and S. Trinajstić, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393–399], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and found the formulas of the F-index, hyper-Zagreb index, and Sigma index for double join of graphs based on the total graph and double corona of graphs based on the total graph. Moreover, some applications of the derived results are also discussed.