Related papers: The star sequence and the general first Zagreb ind…

Starting with the assertion that the problem of isolated, single star formation is essentially solved, this paper examines some of the missing steps needed to go from there to understanding the star formation history of the Universe. Along…

We prove analogues for hypergraphs of Szemer\'edi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemer\'edi theorem of Furstenberg and…

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is…

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…

This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…

The Kolakoski sequence is the unique infinite sequence with values in $\{1,2\}$ and first term $1$ which equals the sequence of run-lengths of itself, we call this $K(1,2).$ We define $K(m,n)$ similarly. A well-known conjecture is that the…

In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of…

Graph inference plays an essential role in machine learning, pattern recognition, and classification. Signal processing based approaches in literature generally assume some variational property of the observed data on the graph. We make a…

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that…

Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities…

In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…

We outline a method to infer the global history of star formation in galaxies with input only from absorption-line observations of quasars. The application of the method to existing data leads to the conclusion that most stars formed at…

The intersection graphs of stars in some tree are known as substar graphs. In this paper we give a characterization of substar graphs by the list of minimal forbidden induced subgraphs. This corrects a flaw in the main result of Chang,…

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for…

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $\Gamma(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(\Gamma(R))$ are adjacent if and only if $x\cdot…

Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…

In this paper, we study the generating functions of multiple $t$-star values with an arbitrary number of blocks of twos, which are based on the results of the corresponding generating functions of multiple $t$-harmonic star sums. These…

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/…

In the paper [I. Gutman, N. Trinajsti\'c, Chem. Phys. Lett. 17 (1972), 535], it was shown that total $\pi$-electron energy ($E$) of a molecule $M$ depends on the quantity $\sum_{v\in V(G)}d_{v}^{2}$ (nowadays known as the "first Zagreb…

For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, \nu}(z)&:= \bigg(2^{a \nu-a+1} a^{-\frac{a(a\nu-a+1)}{2}} \Gamma(a…