On the least size of a graph with a given degree set

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. A theorem of Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] asserts that the least order of a graph with a given degree set D is 1+max(D). We look at the analogous problem concerning the least size of a graph with a given degree set D. We determine the least size for the sets D when (i) |D|?3; (ii) D={1,2,…,n}; and (iii) every element in D is at least |D|. In addition, we give sharp upper and lower bounds in all cases.     

On diameter and inverse degree of a graph

The inverse degree r(G) of a finite graph G=(V,E) is defined as , where is the degree of vertex v. We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erd?s et al. (1988) [5], and by Dankelmann et al. (2008) [4].     

Large bipartite graphs with given degree and diameter

We give constructions of bipartite graphs with maximum Δ, diameter D on B vertices, such that for every D ≥ 2 the lim inf_Δ→∞B. Δ^1-D = b_D > 0. We also improve similar results on ordinary graphs, for example, we prove that lim_Δ→∞N · Δ^?D = 1 if D is 3 or 5. This is a partial answer to a problem of Bollobás.     

Some large graphs with given degree and diameter

This paper considers the (Δ, D) problem: to maximize the order of graphs with given maximum degree Δ and diameter D, of importance for its implications in the design of interconnection networks. Two cubic graphs of diameters 5 and 8 and orders 70 and 286, respectively, and a graph of degree 5, diameter 3 and order 66 are presented, which improve the previously known orders for these values of Δ and D. By its construction, these graphs have a large automorphism group.     

Vertices of given degree in a random graph

This paper concerns the degree sequence d₁ ≥ d₂ ≥ … ≥ d_n of a randomly labeled graph of order n in which the probability of an edge is p(n) ≦ 1/2. Among other results the following questions are answered. What are the values of p(n) for which d₁, the maximum degree, is the same for almost every graph? For what values of p(n) is it true that d₂ > d₂ for almost every graph, that is, there is a unique vertex of maximum degree? The answers are (essentially) p(n) = o(logn/n/n) and p(n)n/logn → ∞. Also included is a detailed study of the distribution of degrees when 0 < lim n p(n)/log n ≦ lim n p(n)/log n < ∞.     

Every connected regular graph of even degree is a Schreier coset graph

Using Petersen's theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.     

Large graphs with given degree and diameter. II

The following problem arises in the study of interconnection networks: find graphs of given maximum degree and diameter having the maximum number of vertices. Constructions based on a new product of graphs, which enable us to construct graphs of given maximum degree and diameter, having a great number of vertices from smaller ones are given; therefore the best values known before are improved considerably.     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence D, satisfying certain conditions. Molloy and Reed defined a parameter Q = Q(D) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R( \begin{align*}\mathcal {D}\end{align*}) and prove that if |Q| = O(n^‐1/3R^2/3) then, with high probability, the size of the largest component of the random graph will be of order Θ(n^2/3R^‐1/3). If |Q| is asymptotically larger than n^‐1/3R^2/3 then the size of the largest component is asymptotically smaller or larger than n^2/3R^‐1/3. Thus, we establish that the scaling window is |Q| = O(n^‐1/3R^2/3). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Large bipartite Cayley graphs of given degree and diameter

Let BC_d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC_d,k≥2(k−1)((d−4)/3)^k−1 for any d≥6 and any even k≥4, and BC_d,k≥(k−1)((d−2)/3)^k−1 for d≥6 and k≥7 such that k≡3 (mod 4).     

Some properties of planar polynomial systems of even degree

Summary We study autonomous systems in the plane of polynomial type. We obtain conditions for the existence of unbounded trajectories of such systems. As a consequence we prove that it does not exist a planar polynomial system of even degree with a global center.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.