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On Graph-Based Cryptography and Symbolic Computations

V. A., Ustimenko (2007)

Serdica Journal of Computing

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 the adjacent eccentric distance sum of graphs

Halina Bielak, Katarzyna Wolska (2015)

Annales UMCS, Mathematica

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 sum of powers of Laplacian eigenvalues of bipartite graphs

Bo Zhou, Aleksandar Ilić (2010)

Czechoslovak Mathematical Journal

For a bipartite graph G and a non-zero real α , we give bounds for the sum of the α th powers of the Laplacian eigenvalues of G 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.

One-two descriptor of graphs

K. CH. Das, I. Gutman, D. Vukičević (2011)

Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques

Ordering the non-starlike trees with large reverse Wiener indices

Shuxian Li, Bo Zhou (2012)

Czechoslovak Mathematical Journal

The reverse Wiener index of a connected graph G is defined as Λ ( G ) = 1 2 n ( n - 1 ) d - W ( G ) , where n is the number of vertices, d is the diameter, and W ( G ) is the Wiener index (the sum of distances between all unordered pairs of vertices) of G . We determine the n -vertex non-starlike trees with the first four largest reverse Wiener indices for n ≥ 8 , and the n -vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for n ≥ 10 .

Parity vertex colorings of binomial trees

Petr Gregor, Riste Škrekovski (2012)

Discussiones Mathematicae Graph Theory

We show for every k ≥ 1 that the binomial tree of order 3k has a vertex-coloring with 2k+1 colors such that every path contains some color odd number of times. This disproves a conjecture from [1] asserting that for every tree T the minimal number of colors in a such coloring of T is at least the vertex ranking number of T minus one.

Parity vertex colouring of graphs

Piotr Borowiecki, Kristína Budajová, Stanislav Jendrol', Stanislav Krajci (2011)

Discussiones Mathematicae Graph Theory

A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χₚ(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χₚ(G) ≤ |V(G)|-α(G)+1, where χ(G) and α(G) are the chromatic number and the independence number of G, respectively. The bounds are improved for trees. Namely, if T is a tree with diameter diam(T) and radius rad(T),...

Partitions of k -branching trees and the reaping number of Boolean algebras

Claude Laflamme (1993)

Commentationes Mathematicae Universitatis Carolinae

The reaping number 𝔯 m , n ( 𝔹 ) of a Boolean algebra 𝔹 is defined as the minimum size of a subset 𝒜 ⊆ 𝔹 ∖ { 𝐎 } such that for each m -partition 𝒫 of unity, some member of 𝒜 meets less than n elements of 𝒫 . We show that for each 𝔹 , 𝔯 m , n ( 𝔹 ) = 𝔯 ⌈ m n - 1 ⌉ , 2 ( 𝔹 ) as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every k -branching tree whose maximal nodes are coloured with ℓ colours contains an m -branching subtree using at most n colours if and only if ⌈ ℓ n ⌉ < ⌈ k m - 1 ⌉ .

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