Extremality of degree-based graph entropies

A nanotube is a nanometer–scale tube-like structure, it is a kind of nanoparticle, and may be large enough to serve as a pipe through which other nanoparticles can be channeled, or, depending on the material, may be used as an electrical conductor or an electrical insulator. For computing the structural information of nanotubes, the graph entropies have become the information theoretic quantities. The graph entropy measure has attracted the research community due to its potential application in discrete mathematics, biology, and chemistry. In this paper, our contribution is to explore graph entropies for structures of some nanotubes based on novel information function, which is the number of different degree vertices along with the number of edges between various degree vertices. More precisely, we computed entropies of some classes of nanotubes such as titania nanotube TNT[Formula: see text], TNT[Formula: see text] and carbon nanotubes HAC[Formula: see text] by making a relation of degree-based topological indices with the help of information function.

This article examines graph entropy measures that depend on the number of dominating and power-dominating sets. To quantify the structural complexity of a graph structure, one uses graph entropies. It is easy to compute these properties for smaller networks, and if reliable approximations are developed, similar metrics can also be used for larger graphs. Using various graph invariants, many graph entropy measures have already been established and computed. So, in this work, a new graph entropy measure, namely, power domination entropy, using the power domination polynomial, is introduced. The domination and power domination polynomials of graphs are used to determine the number of dominating and power dominating sets. Let D(G,ξ) represent the collection of all dominating sets of G with size ξ, dξ(G)=|D(G,ξ)|, and γs be the total number of dominating sets of G. Then, the domination entropy of G with n nodes is defined as Idom(G)=−∑ξ=1ndξ(G)γs(G) log(dξ(G)γs(G)). The domination and power domination entropies for a few graphs are further computed. Following that, a comparison between the domination and power domination entropies of several graphs is provided.

In a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.

Riordan graphs II: Spectral properties

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of vertices, number of edges, maximum vertex degree, minimum vertex degree, the total eccentricity index the first Zagreb indices and the second multiplicative Zagreb index.

From social networks to machine learning to biological network analysis, graphs spring up in all sorts of applications related to either the natural sciences or systems created by humans. While individual nodes and edges express relationships between interacting entities, there is often far greater insight in understanding a graph's larger structural properties, e.g., treewidths, the maximal degree, or the second smallest eigenvalue of an associated graph Laplacian. As these fundamental structural properties are independent of a specific node labeling, it is then worthwhile to examine graph invariants, functions of graphs that do not depend on specific node labels. The paper by Chandrasekaran, Parrilo, and Willsky explores the realm of convex graph invariants, convex functions of a graph's adjacency matrix which are also graph invariants. These convex graph invariants arise in many practical situations and are advantageous in that one can leverage convex optimization ideas and tools to provide tractable algorithms. Three canonical graph structural problems are introduced: graph deconvolution, graph generation, and graph hypothesis testing. Deconvolution seeks to unravel a graph into simpler graph structures. Network analysis in areas such as biological sciences and social networks often rely on this type of decomposition. Graph generation seeks to produce graphs satisfying specified constraints (e.g., each node has a certain degree or the corresponding Laplacian matrix has certain spectral properties). This might arise in situations where one wishes to construct an economical network with certain robustness characteristics. Graph hypothesis testing aims to categorize candidate graphs, i.e., deciding whether they fit better within one graph family or another (where families are specified by graph invariants). The paper argues that the key to understanding and computing convex graph invariants lies in being able to first represent them via elementary invariants. This leads to solution methodologies associated with quadratic assignment problems and tractable approximations via optimization techniques such as linear programming and semidefinite programming. Overall, the paper provides a clear view into the world of convex graph invariants. In addition to providing a general framework for computing these invariants, numerous examples introduce readers unfamiliar with this area to the wide range of practical cases where the notions of invariant convex sets and convex graph invariants emerge.

Introduction/purpose: The temperature of a vertex of a graph of the order n is defined as d/(n-d), where d is the vertex degree. The temperature variant of the Sombor index is investigated and several of its properties established. Methods: Combinatorial graph theory is applied. Results: Extremal values and bounds for the temperature Sombor index are obtained. Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.

Quantifying the importance of specific types of higher-order structures in complex networks

Abstract. In this paper, we present Par@Graph, a software toolbox to reconstruct and analyze complex climate networks having a large number of nodes (up to at least 106) and edges (up to at least 1012). The key innovation is an efficient set of parallel software tools designed to leverage the inherited hybrid parallelism in distributed-memory clusters of multi-core machines. The performance of the toolbox is illustrated through networks derived from sea surface height (SSH) data of a global high-resolution ocean model. Less than 8 min are needed on 90 Intel Xeon E5-4650 processors to reconstruct a climate network including the preprocessing and the correlation of 3 × 105 SSH time series, resulting in a weighted graph with the same number of vertices and about 3.2 × 108 edges. In less than 14 min on 30 processors, the resulted graph's degree centrality, strength, connected components, eigenvector centrality, entropy and clustering coefficient metrics were obtained. These results indicate that a complete cycle to construct and analyze a large-scale climate network is available under 22 min Par@Graph therefore facilitates the application of climate network analysis on high-resolution observations and model results, by enabling fast network reconstruct from the calculation of statistical similarities between climate time series. It also enables network analysis at unprecedented scales on a variety of different sizes of input data sets.

Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distribution. Shannon’s entropy theory initiates graph entropies and develops information-theoretic magnitudes for structural computational evidence of organic graphs and complex networks. Graph entropy measurements are valuable in several scientific fields, such as computing, chemistry, biology, and discrete mathematics. In this study, we investigate the entropy of fractal-type networks by considering cycle, complete, and star networks as base graphs using degree-based topological indices.

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property \(\varPhi \). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there exists one graph satisfying \(\varPhi \)? We approach this question by using formal languages for specifying families of graphs. In particular, we show that certain graph properties can be decided by studying the syntactic monoid of the specification language.

Some forbidden combinations of branches in minimal-ABC trees

The first and second Zagreb indices of a graph are equal‎, ‎respectively‎, ‎to‎ ‎the sum of squares of the vertex degrees‎, ‎and the sum of the products of‎ ‎the degrees of pairs of adjacent vertices‎. ‎We now consider analogous‎ ‎graph invariants‎, ‎based on the second degrees of vertices (number of‎ ‎their second neighbors)‎, ‎called leap Zagreb indices‎. ‎A number of their‎ ‎basic properties is established‎.

Distance-balanced graphs: Symmetry conditions