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Spectral Graph Theory, Topological Indices of Graph, and Entropy

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 10716

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Prof. Dr. Kinkar Chandra Das

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Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Interests: spectral graph theory; algebraic graph theory; topological indices of graphs
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Dear Colleagues,

Spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M theory. Frequently used graph matrices are the adjacency matrix, Laplacian matrix, distance matrix, distance Laplacian matrix distance signless Laplacian matrix, etc. Other graph matrices are used as well. Spectral graph theory includes all these particular theories together with interaction tools. Investigations in the field of the theory of graph spectra include the following topics: extremal problems with some eigenvalue; integral graphs; bounds on eigenvalues; graph invariants that can be used as structure descriptors (in chemistry and elsewhere); study of the energy of a graph and other spectrally based invariants; limit points for specified eigenvalues within some classes of graphs; perturbation problems related to the largest (least) eigenvalue; determination of graphs by various graph spectra; etc.

Molecular descriptors play a significant role in mathematical chemistry, especially in QSPR/QSAR investigations. A topological index is a real number related to a graph that must be a structural invariant. Several topological indices have been defined, and many of them have found applications as means to model chemical, pharmaceutical, and other properties of molecules. The popular topological indices are the Wiener index, Randić index, Zagreb indices, ABC index, GA index, Merrifield–Simmons index, etc.

Graph entropy measures have played an important role in a variety of fields, including information theory, biology, chemistry, and sociology. The entropy of a probability distribution can be interpreted not only as a measure of uncertainty, but also as a measure of information, and the entropy of a graph is an information-theoretic quantity for measuring the complexity of a graph. Information-theoretic network complexity measures have already been intensively used in mathematical and medicinal chemistry, including drug design. So far, numerous such measures have been developed such that it is meaningful to show relatedness between them. Note that several graph entropies have been used extensively to characterize the topology of networks.

This Special Issue aims to offer an opportunity to researchers to share their original work and ideas in investigating various spectral graph theory problems and topological indices with graph entropy. Original research and review articles are welcome. Potential topics include but are not limited to:

Algebraic graph theory;
Topological indices of graphs;
Structural graph theory;
Pure graph theory;
Energy of graphs;
Combinatorics;
Discrete mathematics;
Extremal graph theory.

Prof. Dr. Kinkar Chandra Das
Guest Editor

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Related Special Issue

Spectral Graph Theory, Topological Indices of Graph, and Entropy: Second Edition in Entropy

Published Papers (5 papers)

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Research

11 pages, 1562 KiB

Open AccessFeature PaperArticle

Degree-Based Graph Entropy in Structure–Property Modeling

by Sourav Mondal and Kinkar Chandra Das

Entropy 2023, 25(7), 1092; https://doi.org/10.3390/e25071092 - 21 Jul 2023

Cited by 8 | Viewed by 2194

Abstract

Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose

d_{G} (v_{i})

is degree of the vertex

v_{i}

for each [...] Read more.

Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose

d_{G} (v_{i})

is degree of the vertex

v_{i}

for each

i = 1, 2, \dots, n

. Now, the k-th degree-based graph entropy for G is defined as

I_{d, k} (G) = - \sum_{i = 1}^{n} (\frac{d_{G} {(v_{i})}^{k}}{\sum_{j = 1}^{n} d_{G} {(v_{j})}^{k}} l o g \frac{d_{G} {(v_{i})}^{k}}{\sum_{j = 1}^{n} d_{G} {(v_{j})}^{k}}),

where k is real number. The first-degree-based entropy is generated for

k = 1

, which has been well nurtured in last few years. As

\sum_{j = 1}^{n} d_{G} {(v_{j})}^{k}

yields the well-known graph invariant first Zagreb index, the

I_{d, k}

for

k = 2

is worthy of investigation. We call this graph entropy as the second-degree-based entropy. The present work aims to investigate the role of

I_{d, 2}

in structure property modeling of molecules. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)

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13 pages, 574 KiB

Open AccessArticle

Deeper Exploiting Graph Structure Information by Discrete Ricci Curvature in a Graph Transformer

by Xin Lai, Yang Liu, Rui Qian, Yong Lin and Qiwei Ye

Entropy 2023, 25(6), 885; https://doi.org/10.3390/e25060885 - 1 Jun 2023

Cited by 3 | Viewed by 1709

Abstract

Graph-structured data, operating as an abstraction of data containing nodes and interactions between nodes, is pervasive in the real world. There are numerous ways dedicated to extract graph structure information explicitly or implicitly, but whether it has been adequately exploited remains an unanswered question. This work goes deeper by heuristically incorporating a geometric descriptor, the discrete Ricci curvature (DRC), in order to uncover more graph structure information. We present a curvature-based topology-aware graph transformer, termed Curvphormer. This work expands the expressiveness by using a more illuminating geometric descriptor to quantify the connections within graphs in modern models and to extract the desired structure information, such as the inherent community structure in graphs with homogeneous information. We conduct extensive experiments on a variety of scaled datasets, including PCQM4M-LSC, ZINC, and MolHIV, and obtain a remarkable performance gain on various graph-level tasks and fine-tuned tasks. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)

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Figure 1

17 pages, 872 KiB

Open AccessArticle

Graphic Groups, Graph Homomorphisms, and Graphic Group Lattices in Asymmetric Topology Cryptography

by Meimei Zhao, Hongyu Wang and Bing Yao

Entropy 2023, 25(5), 720; https://doi.org/10.3390/e25050720 - 26 Apr 2023

Cited by 2 | Viewed by 1328

Abstract

Using asymmetric topology cryptography to encrypt networks on the basis of topology coding is a new topic of cryptography, which consists of two major elements, i.e., topological structures and mathematical constraints. The topological signature of asymmetric topology cryptography is stored in the computer by matrices that can produce number-based strings for application. By means of algebra, we introduce every-zero mixed graphic groups, graphic lattices, and various graph-type homomorphisms and graphic lattices based on mixed graphic groups into cloud computing technology. The whole network encryption will be realized by various graphic groups. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)

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Figure 1

18 pages, 1919 KiB

Open AccessArticle

TREPH: A Plug-In Topological Layer for Graph Neural Networks

by Xue Ye, Fang Sun and Shiming Xiang

Entropy 2023, 25(2), 331; https://doi.org/10.3390/e25020331 - 10 Feb 2023

Cited by 2 | Viewed by 2754

Abstract

Topological Data Analysis (TDA) is an approach to analyzing the shape of data using techniques from algebraic topology. The staple of TDA is Persistent Homology (PH). Recent years have seen a trend of combining PH and Graph Neural Networks (GNNs) in an end-to-end manner to capture topological features from graph data. Though effective, these methods are limited by the shortcomings of PH: incomplete topological information and irregular output format. Extended Persistent Homology (EPH), as a variant of PH, addresses these problems elegantly. In this paper, we propose a plug-in topological layer for GNNs, termed Topological Representation with Extended Persistent Homology (TREPH). Taking advantage of the uniformity of EPH, a novel aggregation mechanism is designed to collate topological features of different dimensions to the local positions determining their living processes. The proposed layer is provably differentiable and more expressive than PH-based representations, which in turn is strictly stronger than message-passing GNNs in expressive power. Experiments on real-world graph classification tasks demonstrate the competitiveness of TREPH compared with the state-of-the-art approaches. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)

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11 pages, 275 KiB

Open AccessArticle

Entropy, Graph Homomorphisms, and Dissociation Sets

by Ziyuan Wang, Jianhua Tu and Rongling Lang

Entropy 2023, 25(1), 163; https://doi.org/10.3390/e25010163 - 13 Jan 2023

Cited by 1 | Viewed by 1515

Abstract

Given two graphs G and H, the mapping of

f : V (G) \to V (H)

is called a graph homomorphism from G to H if it maps the adjacent vertices of G to the adjacent vertices of [...] Read more.

Given two graphs G and H, the mapping of

f : V (G) \to V (H)

is called a graph homomorphism from G to H if it maps the adjacent vertices of G to the adjacent vertices of H. For the graph G, a subset of vertices is called a dissociation set of G if it induces a subgraph of G containing no paths of order three, i.e., a subgraph of a maximum degree, which is at most one. Graph homomorphisms and dissociation sets are two generalizations of the concept of independent sets. In this paper, by utilizing an entropy approach, we provide upper bounds on the number of graph homomorphisms from the bipartite graph G to the graph H and the number of dissociation sets in a bipartite graph G. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)

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