Antal Iványi

Energy, Laplacian energy of double graphs and new families of equienergetic graphs

Abstract For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G* is a bipartite graph with bipartition (X, Y), X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj in G. The double graph D[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the neighbors of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs G* and D[G], L-spectra of Gk* the k-th iterated extended double cover of G. We obtain a formula for the number of spanning trees of G*. We also obtain some new families of equienergetic and L-equienergetic graphs.

Construction of infinite de Bruijn arrays

Abstract We construct a periodic array containing every k-ary m × n array as a subarray exactly once. Using the algorithm SUPER (which for k⩾3 generates an infinite k-ary sequence whose beginning parts of length km, m = 1,2,..., are de Bruijn sequences) we also construct infinite km × ∞ k-ary arrays in which each beginning part of size km × kmn − m, n = 1,2,..., as a periodic array, contains every k-ary m × n array exactly once.

Parallel enumeration of degree sequences of simple graphs II

Abstract In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (Gz(n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is Gz(30) = 5 876 236 938 019 300 and on n = 31 vertices is Gz(31) = 22 974 847 474 172 374. Due to Corollary 21 in [52] these results give the number of degree sequences of simple graphs on 30 and 31 vertices.

On vertex independence number of uniform hypergraphs

Abstract Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.

Leader election in synchronous networks

Abstract Worst, best and average number of messages and running time of leader election algorithms of different distributed systems are analyzed. Among others the known characterizations of the expected number of messages for LCR algorithm and of the worst number of messages of Hirschberg-Sinclair algorithm are improved

ON DUMPLING-EATING GIANTS

Publisher Summary This chapter describes the dumpling-eating giants. The analysis of the best case and the worst case of the processing can often be performed using combinatorial considerations. As a common framework for these problems, an animated model is considered in which dwarfs produce infinite sequences of dumplings, and giants (realizing different processing algorithms) eat the dumpings. The units of the eating are bites, and the investigated measure of eating is the average bite size. The appetite of the different giants is characterized by the parameter b. Giant G b is able to eat up most b dumplings of the same sort at one bite. Giant G b chooses for his first bite from the beginning of the first dumpling-sequence so many dumplings, as possible (at most b of the same sort), and he adds to these dumplings so many ones from the beginning of the second, third sequences, as possible.

On partial sorting in restricted rounds

Abstract Let n and k be integers such that n ≥ 2 and 1 ≤ k ≤n. In this paper, we consider the problem of finding an ordered list of the k best players out of n participants by organizing a tournament of rounds of pairwise matches (comparisons). Assuming that (i) in each match there is a winner (no ties) (ii) the relative strength of the players is constant throughout the tournament and (iii) the players’ strengths are transitive, the problem is equivalent to partially sorting n different, comparable objects, allowing parallelization in rounds. The rounds are restricted as one player can only play one match in each round. We propose concrete pairing algorithms and make conjectures about their performance in terms of the worst case number of rounds and matches required. The research article was started by professor Antal Iványi who sadly passed away during the work and was completed in his honor by the co-author. He hopes, in this modest way, to reflect his deep admiration for professor Iványi’s many contributions to the theory, practice and appreciation of algorithm design and analysis.

Tripartite graphs with given degree set

Abstract If k ≥ 1, then the global degree set of a k-partite graph G = (V1, V2, . . . , Vk, E) is the set of the distinct degrees of the vertices of G, while if k ≥ 2, then the distributed degree set of G is the family of the k degree sets of the vertices of the parts of G. We propose algorithms to construct bipartite and tripartite graphs with prescribed global and distributed degree sets consisting from arbitrary nonnegative integers. We also present a review of the similar known results on digraphs.

On the scores and degrees in hypertournaments

Abstract A k-hypertournament H = (V, A), where V is the vertex set and A is an arc set, is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score si(losing score ri) of a vertex is the number of edges containing vi in which vi is not the last element(in which vi is the last element) and the total score of a vertex vi is ti = si − ri. For v ∈ V we denote dH+=∑a∈Hρ(v,a)

Recognition of split-graphic sequences

d_H^ + = \sum\limits_{a \in H} {\rho (v,a)}