We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie i...
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This is a partial list of molecules that contain 10 carbon atoms. Carbon number List of compounds with carbon number 9 List of compounds with carbon number
In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the number modul...
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisi...
Let $r\geq 3$ be an integer and $G$ be a graph. Let $δ(G), Δ(G)$, $α(G)$ and $μ(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper proved $δ(G)α(G)\leq Δ(...
The chromatic number $χ(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$ where $p \in...
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The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: σ(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hur...
In the paper are computed: the number of binary trees with n nodes and k leaves; the number of leaves in the set of all binary trees with n nodes. These are used to compute the number of states in the buddy system....
Actual numbers are highest and lowest number: 100, 200, 300 versus 1, 2, 3, for example. Whereas: the greatest number of people would take great to mean largest. That's the short answer.
Using a bijective proof, we show the number of ways to arrange a maximum number of nonattacking pawns on a $2m\times 2m$ chessboard is ${2m\choose m}^2$, and more generally, the number of ways to arrange a maximum number of nonattacking pawns on a $2...
We examine a family of discrete probability distributions that describes the "spillage number" in the extended balls-in-bins model. The spillage number is defined as the number of balls that occupy their bins minus the total number of occupied bins....
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that
In the paper, the authors establish explicit formulas for the Dowling numbers and their generalizations in terms of generalizations of the Lah numbers and the Stirling numbers of the second kind. These results gen- eralize the Qi formula for the Bell...
Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to define primori...
The number 2 is the second natural number, after 1. Each natural number, including 2, is constructed by succession, that is, by adding 1 to the previous natural number.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. Humans, and many other animals, have 5 digits on their limbs.
In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of compo...