Edge-isoperimetric Problem Research Articles

A new approach to Macaulay posets

We develop a new approach for establishing the Macaulayness of posets representable as cartesian powers of other posets. This approach is based on a problem of constructing an ideal of maximum rank in a poset. Using the relations between the maximum rank ideal problem and the edge-isoperimetric problem on graphs we demonstrate an application of our approach to specification of all posets with a special Macaulay order. We also present a new general construction for additive Macaulay posets and introduce several new families of Macaulay posets.

Edge-isoperimetric problems for cartesian powers of regular graphs

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best possible edge-isoperimetric inequalities for the cartesian powers of arbitrary regular, resp. regular bipartite, graphs with a high density.

The edge-isoperimetric problem on the 600-vertex regular solid

Between 1963 and 1979 the edge-isoperimetric problem was solved on the graphs of all regular convex polytopes except the four-dimensional one with 600 vertices. We present a solution of that last remaining problem, based on a combination of theory and computer calculation.

The edge-isoperimetric problem for discrete tori

The edge-isoperimetric problem has long been solved for cartesian powers of the cycles C 3 and C 4, for which the lexicographic order is the optimal order, and powers of the cycles C n with n>5, which do not have nested optimal subsets. For powers of C 5, it is clear that the lexicographic order is not optimal. We present a solution to the edge-isoperimetric problem for powers of C 5 in the form of an optimal order for the vertices. We then prove that discrete tori of the forms C 5 i × C 4 j × C 3 k and C n × C 5 i × C 4 j × C 3 k have nested optimal subsets for n>5, i,j,k⩾0 , and give an optimal order for members of that class. We conjecture that these are the only discrete tori which have nested optimal subsets.

Morphisms for the Maximum Weight Ideal Problem

A notion of morphism for the Maximum Weight Ideal Problem is defined and applied to solve the Edge–Isoperimetric Problem on the graphs of regular solids and their products.

An Edge-Isoperimetric Problem for Powers of the Petersen Graph

In this paper we introduce a new order on the set of n-dimensional tuples and prove that this order preserves nestedness in the edge isoperimetric problem for the graph P n , defined as the nth cartesian power of the well-known Petersen graph. The cutwidth and wirelength of P n are also derived. These results are then generalized for the cartesian product of P n and the m-dimensional binary hypercube.

On an equivalence in discrete extremal problems

We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edge-isoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quite independently. We present close connections between them. In particular, we show that a number of known results concerning the edge-isoperimetric problem for concrete families of graphs are direct consequences of the Macauleyness of appropriate posets.

General Edge-isoperimetric Inequalities, Part I: Information-theoretical Methods

In combinatorics we often meet two kinds of extremal problems. In one kind, optimal configurations consist of ‘objects’, which are somehow uniformlyspreadin the space under consideration; and in the other kind, optimal configurations consist of ‘objects’, which are somehowcompressed.To the first kind belong packing, covering and coding problems, whereas diametric (especially of Erdös-Ko-Rado type), vertex- and edge-isoperimetric problems belong to the second kind.

Eigenvalues and separation in graphs

Consider a one-to-one map f:V(G)→V(H) of Graphs G and H, where |G|⩽|H|. We let |f|=min{distH(f(x), f(y)):xyεE(G)}, where distH denotes distance in H. Now define sep(G,H), the separation of G into H, to be the maximum of |f| over all such maps f. Using the Kronecker product of matrices, we develop a method for computing, in certain favorable cases, the set of eigenvalues of graphs of the form (G×H)(k). Here G×H refers to the usual graph product, and S(k) (for a graph S) is the graph obtained from S by joining two points of V(S) by an edge if and only if they are at distance at most k in S. Let Q(n) and Cn× Cn denote the n-dimensional cube and the n × n “discrete torus” respectively, and λmin(S) the smallest eigenvalue of a graph S. We apply our method to analyze λmin(Q(n)(k)) and λmin((Cn× Cn)(k)), obtaining exact values for certain k and asymptotically optimal lower bounds for others. Combining these results with one of the results of Alon and Milman, we obtain bounds for the edge isoperimetric problem in the graphs Q(n)(k)c and (Cn× Cn)(k)c, where Sc denotes the graph obtained from a graph S by joining two vertices if and only if they are not joined in S. As a corollary we obtain functions b(k, p)[c(k, p)] such that if a graph G on p points and q edges satisfies q > b(k, p)[q > c(k, p)], then sep(G, Q(n))⩽ k [sep(G, Cn × Cn⩽ k].

Optimal Assignments of Numbers to Vertices

Previous article Next article Optimal Assignments of Numbers to VerticesL. H. HarperL. H. Harperhttps://doi.org/10.1137/0112012PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] William H. Kautz, Optimized data encoding for digital computers, Convention Record of the I. R. E., 2 (1954), 47–57 MR0078064 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi FunctionsVsevolod F. Lev22 December 2015 | SIAM Journal on Discrete Mathematics, Vol. 29, No. 4AbstractPDF (349 KB)Extractors for Circuit SourcesEmanuele Viola29 April 2014 | SIAM Journal on Computing, Vol. 43, No. 2AbstractPDF (271 KB)k-Robust Single-Message TransmissionAndré Kündgen, Michael J. Pelsmajer, and Radhika Ramamurthi31 March 2010 | SIAM Journal on Discrete Mathematics, Vol. 24, No. 1AbstractPDF (327 KB)Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices Based on Semidefinite ProgrammingHans Mittelmann and Jiming Peng2 December 2010 | SIAM Journal on Optimization, Vol. 20, No. 6AbstractPDF (251 KB)Extremal Sets Minimizing Dimension-Normalized Boundary in Hamming GraphsM. Cemil Azizouglu and Ömer Eugeciouglu1 August 2006 | SIAM Journal on Discrete Mathematics, Vol. 17, No. 2AbstractPDF (402 KB)Edge-Bandwidth of GraphsTao Jiang, Dhruv Mubayi, Aditya Shastri, and Douglas B. West1 August 2006 | SIAM Journal on Discrete Mathematics, Vol. 12, No. 3AbstractPDF (282 KB)Optimal Numberings of an $N \times N$ ArrayGraeme Mitchison and Richard Durbin17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 4AbstractPDF (1219 KB)Upper and Lower Bounds on the Complexity of the Min-Cut Linear Arrangement Problem on TreesThomas Lengauer31 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 1AbstractPDF (1346 KB)Complexity Results for Bandwidth MinimizationM. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 34, No. 3AbstractPDF (2269 KB)Maximally Connected Arrays on the n-CubeA. J. Bernstein17 February 2012 | SIAM Journal on Applied Mathematics, Vol. 15, No. 6AbstractPDF (424 KB)Optimal Binary Coding of Ordered NumbersK. Steiglitz and A. J. Bernstein13 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 2AbstractPDF (246 KB) Volume 12, Issue 1| 1964Journal of the Society for Industrial and Applied Mathematics History Submitted:12 February 1963Published online:13 July 2006 InformationCopyright © 1964 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0112012Article page range:pp. 131-135ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics