The Laplacian of a Graph

Developments in graph colouring theory were motivated by the four-colour problem and Heawood's theorem. Both of these were originally formulated as map-colouring problems that can be expressed as colouring graphs embedded on surfaces. This chapter gives an overview of the abundance of results concerning the chromatic number of graphs that are embedded on surfaces . Introduction In 1852 Francis Guthrie asked whether the regions of every planar map can be coloured with four colours in such a way that no two regions of the map with common boundary receive the same colour. In effect, by duality Guthrie was asking whether every planar graph is 4-colourable. This easily stated problem became known as the famous four-colour problem. Attempts to solve it led to many important results in graph theory, but the problem itself remained unsolved for more than a century. It was finally answered in the positive by Appel and Haken [9], [10], [12]. More about its proof comes later in this chapter. For more than a century, the four-colour problem was one of the driving forces that led to developments in graph theory, and specifically graph colouring theory. Its generalization – the Heawood problem – was the main motivation for developments in topological graph theory (see [51] or [49]). Both of these were originally formulated as map-colouring problems, asking about colouring the faces of an embedded graph so that adjacent faces receive different colours. Every map-colouring problem can be expressed as a graph-colouring problem by considering the dual graph of the map. Its vertices correspond to the faces of the map, and two vertices are adjacent if the corresponding faces are adjacent on the surface. In this chapter we discuss the colouring of graphs embedded on surfaces. We start by describing some of the ideas in the proof of the four-colour theorem and give a proof of the list-colouring version of the five-colour theorem that is due to Thomassen.

Spectral Graph Theory is the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of systems has received much interest in the last 15 years. Both algorithm design theory and practical implementations have made substantial progress. There is also a growing number of problems that can be efficiently solved using SDD solvers including: image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics. All these examples can be viewed as special case of convex optimization problems.

Graph theoretical studies have been designed to investigate network topologies during life. Network science and graph theory methods may contribute to a better understanding of brain function, both normal and abnormal, throughout developmental stages. The degree to which childhood epilepsies exert a significant effect on brain network organisation and cognition remains unclear. The hypothesis suggests that the formation of abnormal networks associated with epileptogenesis early in life causes a disruption in normal brain network development and cognition, reflecting abnormalities in later life. Neurological diseases with onset during critical stages of brain maturation, including childhood epilepsy, may threaten this orderly neurodevelopmental process. According to the hypothesis that the formation of abnormal networks associated with epileptogenesis in early life causes a disruption in normal brain network development, it is then mandatory to perform a proper examination of children with new-onset epilepsy early in the disease course and a deep study of their brain network organisation over time. In regards, graph theoretical analysis could add more information. In order to facilitate further development of graph theory in childhood, we performed a systematic review to describe its application in functional dynamic connectivity using electroencephalographic (EEG) analysis, focussing on paediatric epilepsy.

Over the last several years there have been major breakthroughs in the design of approximation algorithms for such classic problems as finding the maximum flow in a graph. Maximum flow for undirected graphs can now be approximately solved in almost linear time. This result by researchers at Berkeley and MIT, I claim, is only the beginning of a new era in efficient algorithm design.Graph theoretic optimization problems, that have been dormant for fifty years are now seeing new and exciting algorithms. These advances have been made possible by Spectral Graph Theory, the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of linear systems has received substantial interest in the last 15 years. Both algorithm design theory and practical implementations have made major progress. Surprisingly, there is an ever growing list of problems that can be efficiently solved using SDD solvers including image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics. All these examples can be viewed as special cases of convex optimization problems that arise from graph problems. I can imagine a world where such optimization problems that seem to require at least quadratic work will all be solvable by practical algorithms guaranteed to run in near linear work and are very parallel.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

A hypergraph is a generalization of a graph in which edges—called hyperedges—can connect any number of vertices, not just two. A superhypergraph further extends this framework by introducing recursive powerset constructions, enabling the representation of hierarchical relationships among hyperedges themselves. A topological graph is a graph embedded in the plane, where each vertex is represented by a distinct point and each edge is drawn as a continuous curve connecting its endpoints. A topological hypergraph extends this idea by representing hyperedges as closed curves that enclose sets of vertices on the plane [13]. In this paper, we introduce the concept of a Topological n-SuperHypergraph, which unifies the structural hierarchy of n-superhypergraphs with the geometric intuition of topological hypergraphs. We anticipate that this new formulation will contribute to future developments in graph theory, topology, and network theory.

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

Networks with a prescribed power-law scaling in the spectrum of the graph Laplacian can be generated by evolutionary optimization. The Laplacian spectrum encodes the dynamical behavior of many important processes. Here, the networks are evolved to exhibit subdiffusive dynamics. Under the additional constraint of degree-regularity, the evolved networks display an abundance of symmetric motifs arranged into loops and long linear segments. Exploiting results from algebraic graph theory on symmetric networks, we find the underlying backbone structures and how they contribute to the spectrum. The resulting coarse-grained networks provide an intuitive view of how the anomalous diffusive properties can be realized in the evolved structures.

There are numerous applications of graph theory and algebraic graph theory in combinatorial optimization and optimal structural analysis. In this paper, a new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented. Furthermore, the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied. This block diagonalization leads to an efficient method for the eigensolution of adjacency and Laplacian matrices of special graphs. The eigenvalues and eigenvectors are used for efficient nodal ordering and partitioning of large structural models. The present method is far more simple than any existing general approach.

Topological indices are numbers that are applied to a graph and can be used to describe specific graph properties through algebraic structures. Algebraic graph theory is a helpful tool in a range of chemistry domains. Because it helps explain how the different symmetries of molecules and crystals affect their structure and dynamics, it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. A topological index converts the chemical structure into a number and contributes a lot in chemical graph theory. In this article, we compute the Wiener index, Zagreb indexes, Wiener polynomial, Hyper-Wiener index, ABC index and eccentricity-based topological index of a nonzero component union graph from vector space.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

This survey accompanies the Josiah Williard Gibbs Lecture that I gave at the 2016 Joint Mathematics Meetings. It provides an introduction to three topics: algebraic and spectral graph theory, the sparsification of graphs, and the recent resolution of the Kadison–Singer Problem. The first and third are connected by the second.

Cluster consensus of opinion dynamics with stubborn individuals

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

Flocking of a team of Lagrangian agents is investigated in this paper. The motion equation of agents with Lagrangian dynamics and its properties are reviewed. The centrifugal and coriolis effects and gravitational terms in the dynamics of agents have been neglected in previous works on flocking problem. However the dynamics of many dynamical systems which would be the agents in practical applications can be assumed as Lagrangian with the above mentioned terms. The proposed flocking scheme is based on a gradient algorithm in which the collective potential is systematically constructed to cause the velocity of the agents to reach a consensus. It is explained that the suggested algorithm satisfies three rules of flocking demonstrated by Reynolds: cohesion, separation, and alignment. For this purpose the Hamiltonian of a team of agents with Lagrangian dynamics is considered and is treated as the Lyapunov like for the Multi-Agent system. Then considering the properties of the Lagrangian systems and also using the algebraic graph theory, it is shown that the derivative of the Hamiltonian is in order that the Reynolds' rules are satisfied and the conditions under which flocking is achieved are derived. The analytical tools provided here are relayed on algebraic graph theory, matrix theory, and control theory. In simulation results, a group of wheeled mobile robots are used to show the effectiveness of the suggested algorithm.