Cycles, Paths, Connectivity and Diameter in Distance Graphs

In this thesis, we apply spectral graph theory to show the non-existence of an expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger's inequalities and determine when the equalities hold. In order to apply Cheeger's inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family.

In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism.

Star-extremal graphs and the lexicographic product

Let G be a hierarchical network (graph) with vertex set and edge set . The preclusion set of a subnetwork (defined as a smaller network but with the same topological properties as the original one) in G is a subset of such that has no subnetwork . The preclusion number of in G is . Similarly, the edge preclusion set of in G is a subset of such that has no subnetwork . The edge preclusion number of in G is . The preclusion number and edge preclusion number are parameters which measure the robustness of interconnection networks in the event of failures. In this paper, we investigate a class of graphs which are constructed by combining the star graph with the bubble-sort graph, and give some preclusion numbers and edge preclusion numbers for this class of graphs.

Chebyshev polynomials and spanning tree formulas for circulant and related graphs

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n

Graph representations have been widely used in pattern recognition thanks to their powerful representation formalism and rich theoretical background. A number of error-tolerant graph matching algorithms such as graph edit distance have been proposed for computing a distance between two labelled graphs. However, they typically suffer from a high computational complexity, which makes it difficult to apply these matching algorithms in a real scenario. In this paper, we propose an efficient graph distance based on the emerging field of geometric deep learning. Our method employs a message passing neural network to capture the graph structure and learns a metric with a siamese network approach. The performance of the proposed graph distance is validated in two application cases, graph classification and graph retrieval of handwritten words, and shows a promising performance when compared with (approximate) graph edit distance benchmarks.

In order to develop an algorithm, computer theories are essential and often used in a mathematical sequence. Most of them without sufficient mathematics are difficult to understand. A mathematical algorithm is, therefore, a better guide to confirm whether or not a software is correct. This implies that a mathematical knowledge-based algorithm is of great concern. Let $S=\{r_{1}, r_{2}, \cdots r_{\varphi(n)}\}$ be a set of all co-prime residues of a positive integer $n$ . An integer $n > 0$ is termed as totient if the sum of co-prime residues of $n$ is $2^{k}n, k\geq 1$ . A graph $G$ with $V$ and $E$ be the set of vertices and edges of $G$ respectively, then $G$ is said to be a totient graph if there exists a one-one function $h:V\rightarrow N$ , whose induced function $h^{*}:E\rightarrow N$ , defined by $f^{*}(ab)=f(a)f(b)$ assigns a totient number for each edge of $G$ . In this piece of work, we propose new labeling algorithms for several classes of graphs.

A graph G is said to be singular when its adjacency matrix A = A(G) is singular, and circulant when A = A(G) is a circulant matrix. In this study, two classes of circulant graphs are studied, and conditions sufficient for these graphs to be nonsingular are established.

Accordion graphs: Hamiltonicity, matchings and isomorphism with quartic circulants

The resistance distance is widely used in random walk, electronic engineering, and complex networks. One of the main topics in the study of the resistance distance is the computation problem. The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$ . Two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and the corona-edge of the subdivision graph $G_{1}\star G_{2}$ , were defined by Lu and Miao. The adjacency spectrum and the signless Laplacian spectrum of the two new graph operations were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r_{2}$ -regular graph. In this paper, we give the closed-form formulas for the resistance distance and Kirchhoff index of $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ in terms of the resistance distance and Kirchhoff index of the factor graphs.

An actual topic in the graph drawing is the question how to draw two edge sets on the same vertex set, the so-called simultaneous drawing of graphs. The goal is to simultaneously find a nice drawing for both of the sets. It has been found out that only restricted classes of planar graphs can be drawn simultaneously using straight lines and without crossings within the same edge set. In this paper, we negatively answer one of the most often posted open questions namely whether any two trees with the same vertex set can be drawn simultaneously crossing-free in a straight line way.

Bipartite bithreshold graphs

In this thesis we study the number of spanning trees in some classes of graphs. This is made possible by the famous matrix tree theorem established by Kirchhoff in 1847 which states that the number of spanning trees in a finite graph is given by the product of the non-zero eigenvalues of the combinatorial Laplacian of the graph divided by the number of vertices. We adapt techniques derived by Chinta, Jorgenson and Karlsson in 2010 for d-dimensional discrete tori to circulant graphs with first generator equals to 1 and to d-dimensional degenerating discrete tori. They are degenerating in the sense that d-p sides of the tori are tending to infinity at the same rate while the p other sides tend to infinity sublinearly with respect to the d-p sides. Furthermore, the results on d-dimensional discrete tori enable to derive asymptotics for the number of spanning trees on d-dimensional orthotope square lattices. Other results obtained in this thesis concern closed formulas for the number of spanning trees in directed and non-directed circulant graphs where the generators vary, that is, they linearly depend on the number of vertices.