Top Research Papers on Graph Theory

Some new bounds on general connected graphs, molecular trees and triangle-free graphs are given.

DOI: https://doi.org/10.1090/noti2379 For example, in network optimization, one moves packages from supplies at some vertices to demands at others, with costs accrued per package across edges, attempting to do so most cheaply. In network flow, one tries to maximize the amount of material that can move from a source to a sink, subject to the capacities along edges and the conservation of flow at other vertices along the way. In various versions of pursuit and evasion, cops and robbers take turns moving along edges with the cops trying to capture robbers by landing on them; one tries to minimize...

This course aims to introduce students to general issues of graph theory arising in the mathematics/computer science undergraduate courses.

Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, ...

A family of bipartite homogeneous algebraic graphs of large girth over K is introduced formed by graphs with sets of points and lines isomorphic to Kn, n > 1, and cycle indicator ≥ 2n + 2 such that their projective limit is well defined and isomorph to an infinite forest.

A family of bipartite homogeneous algebraic graphs of large girth over K formed by graphs with sets of points and lines isomorphic K, n>1 and cycle indicator ≥ 2n+2 such that their projective limit is well defined and isomorphic to an infinite forest.

Finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies have the Ramanujan property, which means that the eigenvalues of their adjacency matrices are as small as possible.

A Weitzenbock formula for connection Laplacians in this setting is proved and a discrete Yang-Mills functional is defined and studied to study its Euler-Lagrange equations.

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.

This work showcases modern chemical graph theory's utility in Chemists' analysis and model development toolkit, and incorporates recent advancements in computer science and applied mathematics that are propelling chemical graph theory into new domains of chemical study.

Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph X ; it is a finite abelian group whose cardinality is equal to the number of spanning trees of X (Kirchhoff’s Matrix Tree Theorem). A specific type of covering graph, called a derived graph , that is constructed from a voltage graph with voltage group G is the object of interest in this paper. Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow p -subgroups of Jacobians in an infinite v...

This paper presents a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures, primarily on the adjacency matrix of graphs.

Applying graph theory for modeling blockchain networks to evaluate decentralization, security, privacy, scalability and NFT Mapping allows comprehensive analytical insights to guide the development of efficient, resilient decentralized infrastructures.

This paper introduces a comprehensive theory of multivariate information measures for multiplex graphs, and discusses and quantify the concepts of synergy and redundancy in graphs for the first time, and introduces consistent nonparametric estimators for these multivariate graphon information--theoretic measures.

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ −→ rc ( T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k -connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of k th diameter 2 has rainbow k -connection number at most approximately k 2 .

Walton and Welsh proved that if a coloopless regular matroid M does not have a minor in { M ( K 3 , 3 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow. Lai, Li and Poon proved that if M does not have a minor in { M ( K 5 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow. We prove that if a coloopless regular matroid M does not have a minor in { M (( P 10 ) ¯3 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow where ( P 10 ) ¯3 is the graph obtained from the Petersen graph P 10 by contracting 3 edges of a perfect matching. As both M ( K 3 , 3 ) and M ( K 5 ) are contractions o...

Let λ ( G ) denote the smallest number of vertices that can be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. In a recent paper, we proved that if n is the number of vertices of G , k is the maximum degree of G , and t is the number of vertices of degree k , then λ ( G ) ≤ n +( k − 1) t 2 k . We also showed that λ ( G ) ≤ nk +1 if G is a tree. In this paper, we provide a new proof of the first bound and use it to determine the graphs that attain the bound, and we also determine the trees that attain the second bound.

We define and study a special type of hypergraph. A σ -hypergraph H = H ( n, r, q | σ ), where σ is a partition of r , is an r -uniform hypergraph having nq vertices partitioned into n classes of q vertices each. If the classes are denoted by V 1 , V 2 ,..., V n , then a subset K of V ( H ) of size r is an edge if the partition of r formed by the non-zero cardinalities | K ∩ V i | , 1 ≤ i ≤ n , is σ . The non-empty intersections K ∩ V i are called the parts of K , and s ( σ ) denotes the number of parts. We consider various types of cycles in hypergraphs such as Berge cycles and sharp cycles in...

The zero-forcing number, Z ( G ) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Z ( G ) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2 δ − 2 ≤ Z ( G ) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Z ( G ) as a function of the girth and δ ; this conjecture is proved in a few cases and numerical ...

The purpose of this paper is to extend the scope of the Ehrhart theory to periodic graphs. We give sufficient conditions for the growth sequences of periodic graphs to be a quasi-polynomial and to satisfy the reciprocity laws. Furthermore, we apply our theory to determine the growth series in several new examples.

This paper discusses which classes of networks allow for a majority of agents to have the wrong impression about what the majority opinion is, that is, to be in a 'majority illusion'.

We consider two-dimensional N =(2 , 2) supersymmetric gauge theory on discretized Riemann surfaces. We find that the discretized theory can be efficiently described by using graph theory, where the bosonic and fermionic fields are regarded as vectors on a graph and its dual. We first analyze the Abelian theory and identify its spectrum in terms of graph theory. In particular, we show that the fermions have zero modes corresponding to the topology of the graph, which can be understood as kernels of the incidence matrices of the graph and the dual graph. In the continuous theory, a scalar curvature a...

The springer gtm test result, lt author name s gt lt shortened article, and the review of modern graph theory by bla bollobas acm, graph theory an introductory course bela bollsobas.

This work starts by introducing and motivating classical matrices associated with a graph, and then shows how to derive combinatorial properties of a graph from the eigenvalues of these matrices.

The Roman { 2} - domination number, γ R 2 of Chellali, Haynes, Hedetniemi, and McRae, is used to prove that if G is a claw-free graph and H is an arbitrary graph, then γ { 2 } ( G (cid:50) H ) ≥ γR 2 ( G

A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system ˙ θ i = ∑ j ∼ i sin ( θ j − θ j ) , where i ∼ j means vertices i and j are adjacent in the corresponding graph. As a consequence, the existence of linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor in 2015 and a new lower bound for the minimum degree that ensures synchronization is found.

An independent set may not contain both a vertex and one of its neighbours. This basic fact makes the uniform distribution over independent sets rather special. We consider the hard-core model, an essential generalization of the uniform distribution over independent sets. We show how its local analysis yields remarkable insights into the global structure of independent sets in the host graph, in connection with, for instance, Ramsey numbers, graph colourings, and sphere packings.

An outer independent double Italian dominating function on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } for which each vertex x ∈ V ( G ) with f ( x ) ∈ { 0 , 1 } then (cid:80) y ∈ N [ x ] f ( y ) (cid:62) 3 and vertices assigned 0 under f are independent. The outer independent double Italian domination number γ oidI ( G ) is the minimum weight of an outer independent double Italian dominating function of graph G . In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, le...

The (vertex) path-table of a tree 𝑇 contains quantitative information about the paths in 𝑇 . The entry ( 𝑖, 𝑗 ) of this table gives the number of paths of length 𝑗 passing through vertex 𝑣 𝑖 . The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree 𝑇 does not determine 𝑇 uniquely. We shall show that in trees, the number of paths passing through edge 𝑥𝑦 can only be expressed in terms of paths pass...

Let G be a connected graph and let w 1 , · · · w r be a list of vertices. We refer to the choice of a triple ( r ; G ; w 1 , · · · w r ), as a metric selection. Let ρ be the shortest path metric of G . We say that w 1 , · · · w r resolves G (metricly) if the function c : V ( G ) −→ Z r given by

An extremal result about vertex covers, attributed by Hajnal [4] to Erd˝os and Gallai [2], is applied to prove the following: If n , k , and t are integers satisfying n ≥ k ≥ t ≥ 3 and k ≤ 2 t − 2, and G is a graph with the minimum number of edges among graphs on n vertices with the property that every induced subgraph on k vertices contains a complete subgraph on t vertices, then every component of G is complete.

This abstract investigates the ways in which ideas and methods from graph theory which can be applied to database systems, offering a thorough synopsis of their benefits, and outlines potential future directions in the intersection of graph theory and database management.

A selection of the emerging theoretical results on approximation and learning properties of widely used message passing GNNs and higher-order GNNS, focusing on representation, generalization and extrapolation are summarized.

In this article, we introduced a new graph, let C be the pyramid graph with vertex set V(C) and edge set E(C). Pyramid graphs are a powerful tool for data visualization. By combining multiple cycles into a single graph, they provide a clear and concise representation of complex data sets, analyzing hierarchical relationships or illustrating the flow of information, pyramid graphs are an excellent choice for visualizing data. We demonstrate some properties of the newly introduced graph such as chromatic polynomial which has many applications in discreet mathematics, in computer science and stud...

This abstract will focus on their significance, practical uses, and most recent developments of graph theory and algorithms, which offer strong tools for studying and comprehending the complicated linkages and structures of complex systems.

We formulate Aubry–Mather theory for Hamiltonians/Lagrangians defined on graphs, study the structure of minimizing measures, and discuss the relationship with weak KAM theory developed in Siconolfi and Sorrentino (2018 Anal. PDE 1 171–211). Moreover, we describe how to transport and interpret these results on networks.

This paper will explore whether brain networks follow scale-free and small-worldness among other graph theory properties.

This paper provides a new perspective on Geary's c using concepts from spectral graph theory/linear algebraic graph theory and provides three types of representations for it: graph Laplacian representation, graph Fourier transform representation, and Pearson’s correlation coefficient representation.

Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $\mu$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $\lambda$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $\lambda$-invariant of the pullback. This formula is analogous to Kida'...

A theoretical analysis structured through the processes of reasoning that students activate when solving graph theory problems is performed, which might be very helpful to design efficient data collection instruments for empirical studies aiming to analyze students’ thinking in this field of mathematics.

A novel combinatorial approach to study point-variety incidences and unit-distance problem in finite fields, and gives tight bounds for both problems under a similar non-degeneracy assumption, and resolves Zarankiewicz type problems for algebraic graphs.

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in $\mathcal F$. We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families $\mathcal F$ and implies several new and existing results. In particular, whenever $\mathrm{EX}(n,\mathcal F)$ contains the complete bipartite graph $K_{k,n-k}$ (or certain similar graphs) then $\mathrm{SPEX}(n,\mathcal F...

Quantum measurements are inherently probabilistic and quantum theory often forbids to precisely predict the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. These relations are, in many cases, tight, ...

It is shown that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded µ-calculus.

Abstract. We apply the quotient graph theory described by Band, Berkolaiko, Joyner and Liu to particular graphs symmetric with respect to S3 and C3 symmetry groups. We find the quotient graphs for the three-edge star quantum graph with Neumann boundary conditions at the loose ends and three types of coupling conditions at the central vertex (standard, δ and preferred-orientation coupling). These quotient graphs are smaller than the original graph and the direct sum of quotient graph Hamiltonians is unitarily equivalent to the original Hamiltonian.

For a k$$ k $$ ‐vertex graph F$$ F $$ and an n$$ n $$ ‐vertex graph G$$ G $$ , an F$$ F $$ ‐tiling in G$$ G $$ is a collection of vertex‐disjoint copies of F$$ F $$ in G$$ G $$ . For r∈ℕ$$ r\in \mathbb{N} $$ , the r$$ r $$ ‐independence number of G$$ G $$ , denoted αr(G)$$ {\alpha}_r(G) $$ , is the largest size of a Kr$$ {K}_r $$ ‐free set of vertices in G$$ G $$ . In this article, we discuss Ramsey–Turán‐type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal F$$ F $$ ‐...

It is argued that the graph machine learning community needs to shift its attention to developing a balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

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The chapter will deal with graph theory and its application in various financial market decisions and the topological properties of the network of stocks will provide a deeper understanding and a good conclusion to the market structure and connectivity.