Short score certificates for upset tournaments.
Shortness coefficient of cyclically 5-connected cubic planar graphs.
Shortness coefficient of cyclically 5-connected cubic planar graphs. (Short Communication).
Shuffle bialgebras
The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...
Shuffle-invariant total orders.
Sierpiński graphs as spanning subgraphs of Hanoi graphs
Hanoi graphs H pn model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S pn arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S pn embeds as a spanning subgraph into H pn if and only if p is odd or, trivially, if n = 1.
Sign patterns that allow eventual positivity.
Sign patterns that require eventual positivity or require eventual nonnegativity.
Sign patterns that require or allow power-positivity.
Signatura of magic and Latin integer squares: isentropic clans and indexing
The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares. Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance...
Signed 2-domination in caterpillars
A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number and the signed total 2-domination number of a graph are variants of the signed domination number and the signed total domination number . Their values for caterpillars are studied.
Signed and minus domination in bipartite graphs
The paper studies the signed domination number and the minus domination number of the complete bipartite graph .
Signed -matchings in graphs.
Signed Chip Firing Games and symmetric Sandpile Models on the cycles
We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points.
Signed degree sets in signed graphs
The set of distinct signed degrees of the vertices in a signed graph is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.
Signed domination and signed domatic numbers of digraphs
Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number of D. A set of signed dominating functions on D with the property that for each...
Signed domination numbers of directed graphs
The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.
Signed graphs with at most three eigenvalues
We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes.
Signed k-independence in graphs
Let k ≥ 2 be an integer. A function f: V(G) → −1, 1 defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence...
Signed ordered knotlike quandle presentations.