Related papers: Theta Graph Designs
We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, and present a construction of an infinite family of maximal nontraceable graphs that realize this bound.
Hypergraphs are an invaluable tool to understand many hidden patterns in large data sets. Among many ways to represent hypergraph, one useful representation is that of weighted clique expansion. In this paper, we consider this…
In this monography, it is proposed to consider the concepts of spectra of edge cuts and edge cycles of a graph as a basic mathematical structure for solving the problem of graph isomorphism. An edge cut is defined by an edge and the…
We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high…
We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum…
An edge cover of a graph is a set of edges such that every vertex has at least an adjacent edge in it. Previously, approximation algorithm for counting edge covers is only known for 3 regular graphs and it is randomized. We design a very…
The graph isomorphism problem is a main problem which has numerous applications in different fields. Thus, finding an efficient and easy to implement method to discriminate non-isomorphic graphs is valuable. In this paper, a new method is…
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
We introduce a new method for decomposing the edge set of a graph, and use it to replace the Regularity lemma of Szemer\'edi in some graph embedding problems. An algorithmic version is also given.
Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory,…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
We investigate the following generalisation of the 'multiplication table problem' of Erd\H{o}s: given a bipartite graph with $m$ edges, how large is the set of sizes of its induced subgraphs? Erd\H{o}s's problem of estimating the number of…
Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Tur\'{a}n-type…
We prove that if the number of edges does not exceed 7 then the asymptotics of eigenvalues of the Dirichlet problem uniquely determine the shape of the graph.
We characterize the graphs with loops whose degree sequences have no repeated values and find their adjacency spectrum. In the case of simple graphs, such graphs are called anti-regular graphs and are examples of threshold graphs. The…
In a rainbow version of the classical Tur\'an problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which…
The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing…
In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…
For two given nonnegative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G$ is the assignment of labels $\{0,1, \cdots, n\}$ to the edges so that two edges having a common vertex are labeled with difference at least $h$ and two…
An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting…