This paper shows some properties of finite sequences on Go-boards. It also provides the partial correspondence between two ways of decomposition of curves induced by cages. [15] provide the notation and terminology for this paper. We adopt the following rules: i, j, k, n are natural numbers, f is a finite sequence of elements of the carrier of E 2 T , and G is a Go-board. One can prove the foll...

and Applied Analysis 3 Definition 2.1. Given F,G ∈ Mnm and an indeterminate s ∈ F, the matrix pencil sF−G is called regular when m n and det sF −G / 0 where 0 is the zero element of M 1,F . In any other case, the pencil is called singular. In this paper, as we are going to see in the next paragraph, we consider the case that the pencil is singular. The next definition is very important, since t...

The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are d...

recently, the algebraic theory of mv -algebras is intensively studied. in this paper, we extend the concept of derivation of $mv$-algebras and we give someillustrative examples. moreover, as a generalization of derivations of $mv$ -algebraswe introduce the notion of $f$-derivations and $(f; g)$-derivations of $mv$-algebras.also, we investigate some properties of them.

Given a coloring f : V (G) → IN of graph G and any subgraph H ⊂ G we define fs(H) = ∑ v∈V (H) f(v). In particular, we denote fs(G) by S(f). The coloring f is called an ICcoloring if for any integer k ∈ [1, S(f)] there is a connected subgraph H ⊂ G such that fs(H) = k. Also, we define the IC-index of G to be M(G) = max{ S(f) : f is an IC-coloring of G }. In this paper we examine some well-known ...

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, λ a real (or complex) number, and Wλ(G) = ∑ k≥1 d(G, k) k . It is proved that for a partial cube G, Wλ+1(G) = |F|Wλ(G)− ∑ F∈F Wλ(G\F ), where F is the partition of E(G) induced by the Djoković-Winkler relation Θ. This result extends previously known result for trees and implies several relations for distance-bas...

Complex networks abound in physical, biological and social sciences. Quantifying a network's topological structure facilitates network exploration and analysis, and network comparison, clustering and classification. A number of Wiener type indices have recently been incorporated as distance-based descriptors of complex networks, such as the R package QuACN. Wiener type indices are known to depe...

Let ) , ( λ χ G denotes the number of proper vertex colourings of with at most G λ colours. G. Birkhoff , observed in 1912 that ) , ( λ χ G is, for a fixed graph , a polynomial in G λ , which is now called the chromatic polynomial of . More precisely, let G be a simple graph and G N ∈ λ . A mapping } , {1,2, ) ( : λ K → G V f is called a λ -colouring of if whenever the vertices and are adjacent...