相关论文: New large graphs with given degree and diameter

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with…

Erd\H{o}s determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. Erd\H{o}s' work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order…

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our…

We give two lower bounds on the largest order of an arc-transitive graph of diameter two and a given degree.

Given a class of graphs G closed under taking minors, we study the maximum degree \Delta_n of random graphs from G with n vertices. We prove several lower and upper bounds that hold with high probability. Among other results, we find…

Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of the similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are…

This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach…

In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5--17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this…

Intuitively speaking, a bipartite graph is mirror if it can be drawn in the Cartesian plane in such a way that, the vertices of one stable are points in x=0, the vertices of the other stable set are points in x=1, the edges are straight…

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or…

We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this…

Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…

In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana…

We provide a novel method for constructing asymptotics (to arbitrary accuracy) for the number of directed graphs that realize a fixed bidegree sequence $d = a \times b$ with maximum degree $d_{max}=O(S^{\frac{1}{2}-\tau})$ for an…

Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$…

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley…

Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…