27,533 research outputs found
On the s-Hamiltonian index of a graph
In modeling communication networks by graphs, the problem of designing s-fault-tolerant networks becomes the search for s-Hamiltonian graphs. This thesis is a study of the s-Hamiltonian index of a graph G.;A path P of G is called an arc in G if all the internal vertices of P are divalent vertices of G. We define l (G) = max{lcub}m : G has an arc of length m that is not both of length 2 and in a K3{rcub}. We show that if a connected graph G is not a path, a cycle or K1,3, then for a given s, we give the best known bound of the s-Hamiltonian index of the graph
The Hamiltonian index of graphs
The Hamiltonian index of a graph G is defined as h ( G ) = min { m : L m ( G ) is Hamiltonian } . In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44], we constructed a graph H ̃ ( m ) ( G ) from G and prove that if h ( G ) ≥ 2 , then h ( G ) = min{ m : H ̃ ( m ) ( G ) has a spanning Eulerian subgraph }
Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs
We construct several families of genus embeddings of near-complete graphs
using index 2 current graphs. In particular, we give the first known minimum
genus embeddings of certain families of octahedral graphs, solving a
longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle
complements, making partial progress on a problem of White. Index 2 current
graphs are also applied to various cases of the Map Color Theorem, in some
cases yielding significantly simpler solutions, e.g., the nonorientable genus
of . We give a complete description of the method, originally
due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio
Exploring the “Middle Earth” of network spectra via a Gaussian matrix function
We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs
Index theorems for quantum graphs
In geometric analysis, an index theorem relates the difference of the numbers
of solutions of two differential equations to the topological structure of the
manifold or bundle concerned, sometimes using the heat kernels of two
higher-order differential operators as an intermediary. In this paper, the case
of quantum graphs is addressed. A quantum graph is a graph considered as a
(singular) one-dimensional variety and equipped with a second-order
differential Hamiltonian H (a "Laplacian") with suitable conditions at
vertices. For the case of scale-invariant vertex conditions (i.e., conditions
that do not mix the values of functions and of their derivatives), the constant
term of the heat-kernel expansion is shown to be proportional to the trace of
the internal scattering matrix of the graph. This observation is placed into
the index-theory context by factoring the Laplacian into two first-order
operators, H =A*A, and relating the constant term to the index of A. An
independent consideration provides an index formula for any differential
operator on a finite quantum graph in terms of the vertex conditions. It is
found also that the algebraic multiplicity of 0 as a root of the secular
determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe
The generalized hierarchical product of graphs
A generalization of both the hierarchical product and the Cartesian
product of graphs is introduced and some of its properties are
studied. We call it the generalized hierarchical product. In fact,
the obtained graphs turn out to be subgraphs of the Cartesian
product of the corresponding factors. Thus, some well-known
properties of this product, such as a good connectivity, reduced
mean distance, radius and diameter, simple routing algorithms and
some optimal communication protocols, are inherited by the
generalized hierarchical product. Besides some of these properties,
in this paper we study the spectrum, the existence of Hamiltonian
cycles, the chromatic number and index, and the connectivity of the
generalized hierarchical product
- …