Generalized Line Graphs
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Author | : Mohra Abdullah Alqahtani |
Publisher | : |
Total Pages | : 137 |
Release | : 2018 |
Genre | : Graph connectivity |
ISBN | : |
Download Generalized Line Graphs Book in PDF, Epub and Kindle
With every nonempty graph, there are associated many graphs. One of the best known and most studied of these is the line graph L (G) of a graph G, whose vertices are the edges of G and where two vertices of L (G) are adjacent if the corresponding edges of G are adjacent. This concept was implicitly introduced by Whitney in 1932. Over the years, characterizations of graphs that are line graphs have been given, as well as graphs whose line graphs have some specified property. For example, Beineke characterized graphs that are line graphs by forbidding certain graphs that can be subgroups. Sedlacek characterized those graphs whose line graph is planar. Harary and Nash-Williams characterized those graphs whose line graph is Hamiltonian. Chartrand and Wall proved that if G is a connected graph all of whose vertices have degree 3 or more, then, although L(G) may not be Hamiltonian, the line graph of L(G) must be Hamiltonian. Over the years, various generalizations of line graphs have been introduced and studied by many. Among them are Schwenk graphs and k-line graphs introduced in 2015 and 2016 here at Western Michigan University. This study introduces a generalization of line graphs and discusses several well-known structural properties of this class of graphs. Furthermore, it establishes a number of characterizations of connected graphs whose generalized line graphs possess some prescribed graph structure.
Author | : Maciej Marek Sysło |
Publisher | : |
Total Pages | : 24 |
Release | : 1983 |
Genre | : |
ISBN | : |
Download A Charakterization of Generalized Line Graphs Whose Compoements are Also Generalized Line Graphs Book in PDF, Epub and Kindle
Author | : Lowell W. Beineke |
Publisher | : Springer Nature |
Total Pages | : 301 |
Release | : 2021-10-29 |
Genre | : Mathematics |
ISBN | : 303081386X |
Download Line Graphs and Line Digraphs Book in PDF, Epub and Kindle
In the present era dominated by computers, graph theory has come into its own as an area of mathematics, prominent for both its theory and its applications. One of the richest and most studied types of graph structures is that of the line graph, where the focus is more on the edges of a graph than on the vertices. A subject worthy of exploration in itself, line graphs are closely connected to other areas of mathematics and computer science. This book is unique in its extensive coverage of many areas of graph theory applicable to line graphs. The book has three parts. Part I covers line graphs and their properties, while Part II looks at features that apply specifically to directed graphs, and Part III presents generalizations and variations of both line graphs and line digraphs. Line Graphs and Line Digraphs is the first comprehensive monograph on the topic. With minimal prerequisites, the book is accessible to most mathematicians and computer scientists who have had an introduction graph theory, and will be a valuable reference for researchers working in graph theory and related fields.
Author | : Maciej M. Sysło |
Publisher | : |
Total Pages | : 24 |
Release | : 1983 |
Genre | : |
ISBN | : |
Download A characterization of generalized line graphs whose complements are also generalized line graphs Book in PDF, Epub and Kindle
Author | : Dragoš Cvetkovic |
Publisher | : Cambridge University Press |
Total Pages | : 316 |
Release | : 2004-07-22 |
Genre | : Mathematics |
ISBN | : 9780521836630 |
Download Spectral Generalizations of Line Graphs Book in PDF, Epub and Kindle
Introduction -- Forbidden subgraphs -- Root systems -- Regular graphs -- Star complements -- The Maximal exceptional graphs -- Miscellaneous results.
Author | : Dragoš M. Cvetković |
Publisher | : |
Total Pages | : 312 |
Release | : 2014-05-14 |
Genre | : MATHEMATICS |
ISBN | : 9781107363151 |
Download Spectral Generalizations of Line Graphs Book in PDF, Epub and Kindle
Line graphs have the property that their least eigenvalue is greater than, or equal to, -2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. It will be an important resource for all researchers with an interest in algebraic graph theory.
Author | : Maciej Marek Sysło |
Publisher | : |
Total Pages | : 14 |
Release | : 1983 |
Genre | : |
ISBN | : |
Download On the Planarity and Outerplanarity of Generalized Line Graphs, Generalized Middle Graphs and Generalized Total Graphs Book in PDF, Epub and Kindle
Author | : Ching-Shui Cheng |
Publisher | : |
Total Pages | : 22 |
Release | : 1982 |
Genre | : |
ISBN | : |
Download On Regular Generalized Line Graph Designs Book in PDF, Epub and Kindle
A class of efficient designs called regular generalized line graph designs are introduced. This class of designs includes many well-known optimum and efficient designs, e.g., balanced incomplete block designs, group-divisible designs with lambda 2 equal lambda 1 + 1, group-divisible designs with lambda 1 equal lambda 2 + 1 and group size two, triangular designs with lambda 2 equal lambda 1 + 1, L2 designs with lambda 2 equal lambda 1 + 1, etc. The optimality of regular generalized line graph designs is investigated. This uses graph theory as a tool and unifies much of the previous work in the area. (Author).
Author | : Blayne Earl Carroll |
Publisher | : |
Total Pages | : |
Release | : 1995 |
Genre | : |
ISBN | : |
Download Subgraph Transformations Book in PDF, Epub and Kindle
Author | : Khawlah Hamad Alhulwah |
Publisher | : |
Total Pages | : 123 |
Release | : 2017 |
Genre | : Graph theory |
ISBN | : |
Download Structures of Derived Graphs Book in PDF, Epub and Kindle
One of the most familiar derived graphs are line graphs. The line graph L(G) of a graph G is the graph whose vertices are the edges of G where two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. One of the best- known results on the structure of line graphs deals with forbidden subgraphs by Beineke. A characterization of graphs whose line graph is Hamiltonian is due to Harary and Nash-Williams. Iterated line graphs of almost all connected graphs were shown to be Hamiltonian by Chartrand. The girth of a graph G is the length of a smallest cycle of G. An r-regular graph of girth g of minimum order is called a cage. Another class of derived graphs having a connection with cages was introduced by Schwenk. For a graph G having girth 2k + 1, the Schwenk graph G* of G has the set of all (k + 1)-paths as its vertex set where two vertices P and Q are adjacent in G* if and only if P and Q have only an end-vertex in common and the vertices of P and Q induce a (2k + 1)-cycle. In this work, we introduce two new classes of derived graphs, called l -line graphs and Z-graphs. The concept of l -line graphs is a generalization of line graphs and Schwenk graphs, while the Z-graphs provide a different view of certain line graphs. We primarily study the structures of these derived graphs. Results, conjectures and problems on the structural properties such that connectedness, decompositions, Hamiltonicity and planarity of these graphs are presented.