Odd-vertex-degree trees maximizing Wiener index
Offensive alliances in graphs
A set S is an offensive alliance if for every vertex v in its boundary N(S)- S it holds that the majority of vertices in v's closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number...
On a matching distance between rooted phylogenetic trees
The Robinson-Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values...
On bounded paradoxical subsets of the plane
On connections between hypergraphs and algebras
The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove...
On Fully Split Lacunary Polynomials in Finite Fields
We estimate the number of possible degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
On Graph-Based Cryptography and Symbolic Computations
We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large...
On identifying codes in the King grid that are robust against edge deletions.
On partitionable, confidentially connected and unbreakable graphs.
On some inequalities for the skew Laplacian energy of digraphs.
On the adjacent eccentric distance sum of graphs
In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum. Vol. 7 (2O02) no. 26. 1280-1294]. The adjaceni eccentric distance sum index of the graph G is defined as [...] where ε(υ) is the eccentricity of the vertex υ, deg(υ) is the degree of the vertex υ and D(υ) = ∑u∊v(G) d (u,υ)is the sum...
On the dimer problem and the Ising problem in finite 3-dimensional lattices.
On the estrada index of graphs with given number of cut edges.
On the expectation and variance of the reversal distance.
On the sum of powers of Laplacian eigenvalues of bipartite graphs
For a bipartite graph and a non-zero real , we give bounds for the sum of the th powers of the Laplacian eigenvalues of using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.
On the upper bounds for the first Zagreb index
One-two descriptor of graphs
Ordering the non-starlike trees with large reverse Wiener indices
The reverse Wiener index of a connected graph is defined as where is the number of vertices, is the diameter, and is the Wiener index (the sum of distances between all unordered pairs of vertices) of . We determine the -vertex non-starlike trees with the first four largest reverse Wiener indices for , and the -vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for .
Original title unknown
Orthogonal art galleries with holes: a coloring proof of Aggarwal's theorem.