Graph-theoretic Invariants Research Articles

Binomial edge ideals of cographs

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.

Matching Numbers and Dimension of Edge Ideals

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) and ind-match(G) the matching number, minimum matching number and induced matching number of G, respectively. Let $$K[V(G)] = K[x_{1}, \ldots , x_{n}]$$ denote the polynomial ring over a field K and $$I(G) \subset K[V(G)]$$ the edge ideal of G. The relationship between these graph-theoretic invariants and ring-theoretic invariants of the quotient ring K[V(G)]/I(G) has been studied. In the present paper, we study the relationship between match(G), min-match(G), ind-match(G) and $${\hbox {dim}}K[V(G)]/I(G)$$ .

Algebraic Bayesian networks: Checking backbone connectivity

The paper investigates the construction of a joint graph as a global structure of network based on its primary structure, one of the problems arising in machine learning of bases of knowledge patterns with uncertainty, presented in the form of algebraic Bayesian networks. The aim of the research is to propose methods for solving the inverse problem. As the results, algorithms for checking a graph for belonging to a family of joint graphs and a family of minimal joint graphs are proposed, and estimates of their computational complexity are made. An improved version for the special case and an improvement for the general case on average are also proposed for the algorithm for checking membership in a family of joint graphs. The problem of recognition of joint graphs has not been previously researched; issue is being addressed for the first time as currently drafted. The theoretical significance lies in the possibilities for applying the results in further researches of graph-theoretic invariants in the global structures of algebraic Bayesian networks.

Generalized Hamming weights of projective Reed–Muller-type codes over graphs

Let G be a connected graph and let X be the set of projective points defined by the column vectors of the incidence matrix of G over a field K of any characteristic. We determine the generalized Hamming weights of the Reed–Muller-type code over the set X in terms of graph theoretic invariants. As an application to coding theory we show that if G is non-bipartite and K is a finite field of char(K)≠2, then the rth generalized Hamming weight of the linear code generated by the rows of the incidence matrix of G is the rth weak edge biparticity of G. If char(K)=2 or G is bipartite, we prove that the rth generalized Hamming weight of that code is the rth edge connectivity of G.

Graphs with a given diameter that maximise the Wiener index

The Wiener index of a graph is one of the most recognised and very well-researched topological indices, i.e. graph theoretic invariants of molecular graphs. Nonetheless, some interesting questions remain largely unsolved despite being easy to state and comprehend. In this paper, we investigate a long-standing question raised by Plesník in 1984, namely, which graphs with a given diameter d attain the maximum value with respect to the Wiener index. Our approach to the problem is twofold – first we investigate the graphs with diameter smaller than or equal to 4, and then restrict our attention to graphs with diameter equal to n−c for c ≥ 1. Specifically, we provide a complete characterisation of sought-after graphs for 1 ≤ c ≤ 4 and solve the general case for c small enough in comparison to n. Along the way, we state some conjectures and propose an extension to our work.

A Spectral Graph Theoretic Approach for Monitoring Multivariate Time Series Data From Complex Dynamical Processes

The objective of this paper is to monitor complex process dynamics manifest in multivariate (multidimensional) time series data using a spectral (algebraic) graph theoretic approach. We test the hypothesis that the spectral graph-based topological invariants detect incipient process drifts earlier [lower average run length (ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )] and with higher fidelity (consistency of detection) when compared with the conventional statistics-based approaches. The presented approach maps a multidimensional sensor data stream X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N×d</sup> (visualize N as time and d as the number of sensors) as an unweighted and undirected network graph G(V, E), indexed by its vertices V and edges E, i.e., X → G(V, E). The rationale is that the graph-based topological invariants are surrogate representatives of the system state. We compare the monitoring performance of spectral graph theoretic invariants with conventional statistical features in an exponentially weighted moving average control chart setting. The practical utility of the approach is substantiated in the context of process monitoring in two advanced manufacturing scenarios, namely, ultraprecision machining (UPM) and semiconductor chemical mechanical planarization. These studies corroborate the hypothesis that graph theoretic invariants, when used as monitoring statistics, lead to lower ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and more consistent detections in contrast to conventional statistical features. For instance, in the UPM case, the fault detection delay using graph theoretic invariants is less than 160 ms, compared with over 8 s of delay with statistical features.

Spatial “Artistic” Networks: From Deconstructing Integer-Functions to Visual Arts

Deconstructivism is an aesthetically appealing architectonic style. Here, we identify some general characteristics of this style, such as decomposition of the whole into parts, superposition of layers, and conservation of the memory of the whole. Using these attributes, we propose a method to deconstruct functions based on integers. Using this integer-function deconstruction we generate spatial networks which display a few artistic attributes such as (i) biomorphic shapes, (ii) symmetry, and (iii) beauty. In building these networks, the deconstructed integer-functions are used as the coordinates of the nodes in a unit square, which are then joined according to a given connection radius like in random geometric graphs (RGGs). Some graph-theoretic invariants of these networks are calculated and compared with the classical RGGs. We then show how these networks inspire an artist to create artistic compositions using mixed techniques on canvas and on paper. Finally, we call for avoiding that the applicability of (network) sciences should not go in detriment of curiosity-driven, and aesthetic-driven, researches. We claim that the aesthetic of network research, and not only its applicability, would be an attractor for new minds to this field.

Graph theoretic invariants for operator algebras associated to topological dynamics

We expand on some invariants used for classifying nonselfadjoint operator algebras. Specifically to nonselfadjoint operator algebras for which there is a homomorphism (which is also a conditional expectation) onto a commutative diagonal we construct an edge-colored directed graph which can be used as an operator algebra invariant.

Characteristics of Graph Braid Groups

We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.

Immanantal invariants of graphs

Something between an expository note and an extended research problem, this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra. There are a variety of ways to assign a real n× n matrix K( G) to each n-vertex graph G, so that G and H are isomorphic if and only if K( G) and K( H) are permutation similar. It follows that G and H are isomorphic only if K( G) and K( H) are similar, i.e., that similarity invariants of K( G) are graph theoretic invariants of G, an observation that helps to explain the enormous literature on spectral graph theory. The focus of this article is the permutation part, i.e., on matrix functions that are preserved under permutation similarity if not under all similarity.

Alkane Isomer Combinatorics: Stereostructure Enumeration and Graph-Invariant and Molecular−Property Distributions

Combinatoric manipulative methodology to characterize different isomer-class features is extended and illustrated in application to alkane isomers. Stereoisomeric enumerations and various molecular−property characteristics for structural-isomer classes are made. For a range of carbon-atom counts N we enumerate the following: the mean number of diastereomers per N-carbon structural isomer; the mean number of stereoisomers per N-carbon diastereomer; and the mean number of “conformers” per N-carbon stereoisomer. Also we consider distributions of several graph-theoretic properties, characterizing the distributions in terms of averages, variances, covariances, and extremal values. The results for these graph-theoretic invariants are used to predict (via substructural cluster expansion) some molecular properties (heats of formation and magnetic susceptibilities) averaged over these classes. The associated proper standard deviations as well as the extremal values for these cluster-expansion approximants are als...

Characterization of Isospectral Graphs Using Graph Invariants and Derived Orthogonal Parameters

Numerical graph theoretic invariants or topological indices (TIs) and principal components (PCs) derived from TIs have been used in discriminating a set of isospectral graphs. Results show that lower order connectivity and information theoretic TIs suffer from a high degree of redundancy, whereas higher order indices can characterize the graphs reasonably well. On the other hand, PCs derived from the TIs had no redundancy for the set of isospectral graphs studied.

Design of Topological Indices. Part 10. Parameters Based on Electronegativity and Covalent Radius for the Computation of Molecular Graph Descriptors for Heteroatom-Containing Molecules

Two new approaches are presented for the calculation of atom and bond parameters for heteroatom-containing molecules used in computing graph theoretic invariants. In the first approach, the atom and bond weights are computed on the basis of relative atomic electronegativity, using carbon as standard. In the second system, the relative covalent radii are used to compute atom and bond weights, again with the carbon atom as standard. The new definition of the atom and bond parameters leads to a periodic variation versus the atomic number Z, with a more natural variation when compared with the parameters defined only by Z. The two approaches are used to define and compute topological indices based on graph distance. A quantitative structure−property relationship study is reported for boiling points of 185 acyclic compounds with one or two oxygen or sulfur atoms (devoid of hydrogen bonding), in terms of four or five molecular descriptors.

Local monodromy on the fundamental groups of algebraic curves along a degenerate stable curve

We consider the universal local deformation of a stable pointed curve and the associated monodromy representation on the fundamental group of a smooth fiber of this deformation. The image of the monodromy is an abelian subgroup of the outer automorphism of the fundamental group of a smooth fiber, and has a filtration induced from the weight filtration of the fundamental group. By using a graph of groups, we give a combinatorial description of the monodromy representation, and show that the induced filtration is determined by graph-theoretic invariants of the dual graph of the stable pointed curve.

A characterization and hereditary properties for partition graphs

A general partition graph is an intersection graph G on a set S so that for every maximal independent set M of vertices in G, the subsets assigned to the vertices in M partition S. It is shown that such graphs are characterized by there existing a clique cover of G so that every maximal independent set has a vertex from each clique in the cover. A process is described showing how to construct a partition graph with certain prescribed graph theoretic invariants starting with an arbitrary graph with similar invariants. Hereditary properties for various graph products are examined. It is also shown that partition graphs are preserved by removal of any vertex whose closed neighborhood properly contains the closed neighborhood of some other vertex.

On characterization of three-dimensional structures

Molecular descriptors currently used in structure–property–activity studies are based on molecular graphs rather than on a structure as a three-dimensional object. Here we suggest a way to amend graph-theoretic invariants with additional geometric information, thereby providing new molecular descriptors for possible use in quantitative structure–activity correlations. In the approach we assume molecular structures embedded on a regular grid. As an illustration we consider first chains of different length embedded on graphite-like lattice. Subsequently, we consider all possible conformations of hexatriene. Although the cases considered here relate to graphite-like lattice, the approach is general and applies to embedings of molecules on three-dimensional lattices, or, in general, to molecules of arbitrary spatial conformations.

Realizability of p-point graphs with prescribed minimum degree, maximum degree, and point-connectivity

In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a ( p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers ( a, b, c, d) was called ( p, Δ, δ, λ) realizable if there is a ( p, Δ, δ, λ) graph with p = a, Δ = b, δ = c, and λ = d. In this work, we consider the more difficult case of ( p, Δ, δ, κ) realizability, where κ is the point-connectivity. Necessary and sufficient conditions for a quadruple to be ( p, Δ, δ, κ) realizable are derived.