194 research outputs found
On the connectivity of p-diamond-free vertex transitive graphs
AbstractLet G be a graph of order n(G), minimum degree δ(G) and connectivity κ(G). We call the graph G maximally connected when κ(G)=δ(G). The graph G is said to be superconnected if every minimum vertex cut isolates a vertex.For an integer p≥1, we define a p-diamond as the graph with p+2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. Usually, the 1-diamond is triangle and the 2-diamond is diamond. We call a graph p-diamond-free if it contains no p-diamond as a (not necessarily induced) subgraph. A graph is vertex transitive if its automorphism group acts transitively on its vertex set.In this paper, we give some sufficient conditions for vertex transitive graphs to be maximally connected. In addition, superconnected p-diamond-free (1≤p≤3) vertex transitive graphs are characterized
Spectral Fundamentals and Characterizations of Signed Directed Graphs
The spectral properties of signed directed graphs, which may be naturally
obtained by assigning a sign to each edge of a directed graph, have received
substantially less attention than those of their undirected and/or unsigned
counterparts. To represent such signed directed graphs, we use a striking
equivalence to -gain graphs to formulate a Hermitian adjacency
matrix, whose entries are the unit Eisenstein integers Many well-known results, such as (gain) switching and eigenvalue
interlacing, naturally carry over to this paradigm. We show that non-empty
signed directed graphs whose spectra occur uniquely, up to isomorphism, do not
exist, but we provide several infinite families whose spectra occur uniquely up
to switching equivalence. Intermediate results include a classification of all
signed digraphs with rank , and a deep discussion of signed digraphs with
extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues
Oriented paths in n-chromatic digraphs
In this thesis, we try to treat the problem of oriented paths in n-chromatic
digraphs. We first treat the case of antidirected paths in 5-chromatic
digraphs, where we explain El-Sahili's theorem and provide an elementary and
shorter proof of it. We then treat the case of paths with two blocks in
n-chromatic digraphs with n greater than 4, where we explain the two different
approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in
Addario-Berry et al.'s proof and provide a correction for it.Comment: 25 pages, Master thesis in Graph Theory at the Lebanese Universit
The mincut graph of a graph
In this paper we introduce an intersection graph of a graph , with vertex
set the minimum edge-cuts of . We find the minimum cut-set graphs of some
well-known families of graphs and show that every graph is a minimum cut-set
graph, henceforth called a \emph{mincut graph}. Furthermore, we show that
non-isomorphic graphs can have isomorphic mincut graphs and ask the question
whether there are sufficient conditions for two graphs to have isomorphic
mincut graphs. We introduce the -intersection number of a graph , the
smallest number of elements we need in in order to have a family of subsets, such that for each subset. Finally we
investigate the effect of certain graph operations on the mincut graphs of some
families of graphs
A Study of Arc Strong Connectivity of Digraphs
My dissertation research was motivated by Matula and his study of a quantity he called the strength of a graph G, kappa\u27( G) = max{lcub}kappa\u27(H) : H G{rcub}. (Abstract shortened by ProQuest.)
Kings in the Direct Product of Digraphs
A k-king in a digraph D is a vertex that can reach every other vertex in D by a directed path of length at most k. A king is a vertex that is a k-king for some k. We will look at kings in the direct product of digraphs and characterize a relationship between kings in the product and kings in the factors. This is a continuation of a project in which a similar characterization is found for the cartesian product of digraphs, the strong product of digraphs, and the lexicographic product of digraphs
Graph Theory with Applications to Statistical Mechanics
This work will have two parts. The first will be related to various types of graph connectivity, and will consist of some exposition on the work of Andreas Holtkamp on local variants of vertex connectivity and edge connectivity in graphs. The second part will consist of an introduction to the field of physics known as percolation theory, which has to do with infinite connected components in certain types of graphs, which has numerous physical applications, especially in the field of statistical mechanics
Spectral characterizations of complex unit gain graphs
While eigenvalues of graphs are well studied, spectral analysis of complex unit gain graphs is still in its infancy. This thesis considers gain graphs whose gain groups are gradually less and less restricted, with the ultimate goal of classifying gain graphs that are characterized by their spectra. In such cases, the eigenvalues of a gain graph contain sufficient structural information that it might be uniquely (up to certain equivalence relations) constructed when only given its spectrum. First, the first infinite family of directed graphs that is – up to isomorphism – determined by its Hermitian spectrum is obtained. Since the entries of the Hermitian adjacency matrix are complex units, these objects may be thought of as gain graphs with a restricted gain group. It is shown that directed graphs with the desired property are extremely rare. Thereafter, the perspective is generalized to include signs on the edges. By encoding the various edge-vertex incidence relations with sixth roots of unity, the above perspective can again be taken. With an interesting mix of algebraic and combinatorial techniques, all signed directed graphs with degree at most 4 or least multiplicity at most 3 are determined. Subsequently, these characterizations are used to obtain signed directed graphs that are determined by their spectra. Finally, an extensive discussion of complex unit gain graphs in their most general form is offered. After exploring their various notions of symmetry and many interesting ties to complex geometries, gain graphs with exactly two distinct eigenvalues are classified
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