Top Research Papers on Graph Theory

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 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

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.

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

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...

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

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 ...

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.

General Theory: Hypergraphs. Fractional Matching. Fractional Coloring. Fractional Edge Coloring. Fractional Arboricity and Matroid Methods. Fractional Isomorphism. Fractional Odds and Ends. Appendix. Bibliography. Indexes.

This document is a long abstract of my research work, concerning graph theory and algorithms on graphs, that summarizes some results, gives ideas of the proof for some of them and presents the context of the different topics together with some interesting open questions connected to them.

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.

A graph is a set of points (commonly called vertices or nodes) in space that are interconnected by a set of lines (called edges). For a graph G, the edge set is denoted by E and the vertex set by V, so that G= (V, E). Common nomenclature denotes the number of vertices |V| by n and the number of edges |E| by m. Fig. 1 shows a graph G with V= {v1, v2, v3, v4, v5}, E = {e1, e2, e3, e4, e5, e6, e7}, n = 5, and m = 7. If, within E, each edge is specified by its pair of endpoints (e.g. for the example of Fig. 1, e1 is replaced by (v1, v2) etc.), the figure can be dispensed with.

These notes have not been checked by Prof. A.G. Thomason and should not be regarded as ocial notes for the course. In particular, the responsibility for any errors is mine please email Sebastian Pancratz (sfp25) with any comments or corrections.

Fundamental concepts Connectedness Path problems Trees Leaves and lobes The axiom of choice Matching theorems Directed graphs Acyclic graphs Partial order Binary relations and Galois correspondences Connecting paths Dominating sets, covering sets and independent sets Chromatic graphs Groups and graphs Bibliography List of concepts Index of names.

This book is based on Graph Theory courses taught by P.A. Petrosyan, V.V. Mkrtchyan and R.R. Kamalian at Yerevan State University.

Graphs may be classified in many ways apart from their order. A complete graph has each distinct pair of vertices connected by an edge. Many interesting problems arise in looking at complete graphs, several of which have to do with determining whether or not it is possible to pass through every vertex and edge once and only once (a Peterson cycle). A connected graph allows movement from any vertex to another moving along edges of the graph.

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In this unit, you will first determine the area of any given regular polygon and explore what happens to the shape of a polygon as the number of sides increase.

Mathematics acts an important and essential need in different fields. One of the significant roles in mathematics is played by graph theory that is used in structural models and innovative methods, models in various disciplines for better strategic decisions. In mathematics, graph theory is the study through graphs by which the structural relationship studied with a pair wise relationship between different objects. The different types of network theory or models or model of the network are called graphs. These graphs do not form a part of analytical geometry, but they are called graph theory, ...

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.

Some of the conclusions from statistical graph theory are reiterated in the sixth section, but not fully resolved therein, as well as some of the redundancies in the later sections.

This course is intended for second and third year undergraduate students in Mathematics or Mathematics and Computer Science with first year courses in mathematics, most notably Discrete Mathematics.

You are encouraged to work together on solving homework problems, but please put their names clearly at the top of the assignment. Everyone must turn in their own independently written solutions. Homework is due at the beginning of class. def = {0, 1} 2012 , the set of all binary strings of length 2012, in which two vertices are connected if and only if they differ in precisely 1006 coordinates. Prove that G has a perfect matching. 2. Prove that any simple graph G on 2n vertices with δ(G) ≥ n + 10 has at least 12 edge-disjoint perfect matchings. 3. Describe all simple connected graphs for whic...

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...

5 Tightness of the Bounds * 20 5.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 A Look Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.3 Cayley Graphs and Their Spectrum . . . . . . . . . . . . . . . . . . . 27 5.4 The Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.5 The Hypercube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

This article discusses Caley graphs of group with graph theory and discusses isomorphism of Caley graphs of group with isomorphism theory of directed graphs. The article extends the concept of subgraph in graph theory. Furthemore, we discuss the concept and application of Caley graphs of group.

We consider classes of graphs which are easily seen to have many perfect matchings. The class of grid graphs is our main example. We then consider what properties to impose on choosing a subset of vertices A ⊆ V (G) for vertex deletion in a graph G (from such a class) so that the vertex deleted subgraph G− A has a perfect matching. Certain conditions are easy. An even number of vertices must be deleted. If the graph is bipartite then the deleted vertices must have equal numbers from both parts of the bipartition. Also one cannot delete all the neighbours of a given vertex.

The main results of this thesis show that the Kelmans transformation is a very effective tool in many extremal alge- braic graph theoretic problems and attain a breakthrough in a problem of Eva Nosal by the aid of this transformation.

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 brief introduction to design theory and strongly regular graphs, Quasi-symmetric designs, and self-orthogonal codes and designs.

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Let A l Ak be n x n matrices over a commutative ring R with identity. Graph theoretic methods are established to compute the standard polynomial [A , ... I Ak]. It is proved that if k < 2n 2, and if the characteristic of R either is zero or does not divide 4I(V2 n) 2, where I denotes the greatest integer function, then there exist n x n skew-symmetric matrices A 1 . . . , Ak such that [A 1, . . . AkI AO.

This report lists all papers prepared under the sponsorship of the AFOSR grant on topological graph theory and the combinatorial enumeration theory.

SCOT is presented, a system for automatic theory construction in the domain of Graph Theory that takes into account the main processes related to theory construction: concept construction, example production, example analysis, conjecture construction, and conjecture analysis.

Some conclusions about well-founded relations are given taking of the concepts about well-structured graphs.This is happlications of graph theory in set theory.

The aim of the paper is to present characteristics of the role of this theory in agribusiness enterprises and its impact on the agricultural sector.

This book presents a topological approach to combinatorial configurations, in particular graphs, by introducing a new pair of homology and cohomology via polyhedra. On this basis, a number of problems are solved using a new approach, such as the embeddability of a graph on a surface (orientable and nonorientable) with given genus, the Gauss crossing conjecture, the graphicness and cographicness of a matroid, and so forth. Notably, the specific case of embeddability on a surface of genus zero leads to a number of corollaries, including the theorems of Lefschetz (on double coverings), of MacLane...

The authors present a novel graph analytics framework that allows for uncued analysis of very large datasets and combines traditional computer science techniques with signal processing in the context of graph data, creating a new research area at the intersection of the two fields.

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In this book you are invited to the world of the application of the graph theory to chemistry, especially on the problem how the topology of a molecule determines its reactivity toward a specific reaction and how the graphs helps you understand these relationships.

This paper describes the applicability of Game Theory in operation Research using Graph Theoritical tools, which deal with the representation of system structure by means of a connected graph and subsequent analysis through appropriate study of digraph.

This paper builds two graph models whose theories are the set of equations satisfied in, respectively, any graph model and any sensible graph model, and conjecture that the least sensible graph theory is equal to ℋ.

In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned.

Fourier analysis of Boolean functions is applied to solve problems in social choice theory and property testing and examines Arrow’s impossibility theorem and the BLR test.

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With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the information one can deduce from one matrix representation (e.g., the adjacency matrix) is similar to th...

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