In this study we are interested mainly in investigating the relations between two graph irregularity measures which are widely used for structural irregularity characterization of connected graphs. Our study is focused on the comparison and evaluation of the discriminatory ability of irregularity measures called degree deviation S(G) and degree variance Var(G). We establish various upper bounds for irregularity measures S(G) and Var(G). It is shown that the Nikiforov's inequality which is valid for connected graphs can be sharpened in the form of Var(G) < S(G)/2. Among others it is verified that if G is a bidegreed graph then the discrimination ability of S(G) and Var(G) is considered to be completely equivalent.

Let G be a finite simple graph with Laplacian polynomial ψ(G,λ) = ∑ k=0n(−1)n−kck(G)λk. In an earlier paper, we computed the coefficient of cn−4 for trees with respect to some degree-based graph invariant. The aim of this paper is to continue this work by giving an exact formula for the coefficient cn−5 in the polynomial ψ(G,λ). As a consequence of this work, the Laplacian coefficients cn−k, k = 2, 3, 4, 5, for some know trees were computed.

‎Suppose $G$ is a finite non-abelian group and $\Gamma(G)$ is a graph with non-central conjugacy classes of $G$ as its vertex set. Two vertices $L$ and $K$ in $\Gamma(G)$ are adjacent if there are $a \in L$ and $b \in K$ such that $ab = ba$. This graph is called the commuting conjugacy class graph of $G$. The purpose of this paper is to compute the commuting conjugacy class graph of the finite $2-$groups $G_n(m)$ and $G[n]$.

Let G be a graph with vertex set V(G) and edge set E(G). The vertexedge degree of the vertex v, deG(v), equals to the number of different edges that are incident to any vertex from the open neighborhood of v. Also, the edge -vertex degree of the edge e = uv, dvG(e), equals to the number of vertices of the union of the open neighborhood of u and v. In this paper, the vertex -edge connectivity index, Ov, and the edge -vertex connectivity index, Oe, of a graph G were introduced. These are defined as Ov(G) = sigma v is an element of V(G) deG(v)dG(v) and Oe(G) = sigma e=uv is an element of E(G) dG(e)dvG(e), where dG(v) is the degree of a vertex v E V(G) and dG(e) is the number of edges in E(G) that are adjacent to e. In this paper, we will study the main properties of Ov(G), Oe(G) and establish some upper and lower bounds for them. The numbers Ov and Oe for titania nanotubes are also computed.

Let H be a finite abelian group and; ∀h ∈ H⟩ be the generalized dihedral group of H.The aim of this paper is to compute the number of group homomorphisms between two generalized dihedral groups and a generalized dihedral group and an abelian group.One of these results generalized an earlier work by J. W. Johnson published in 2013.

The set of all centralizers of elements in a finite group G is denoted by Cent(G) and G is called n-centralizer if |Cent(G)|=n. In this paper, the structure of centralizers in a non-abelian finite group G with this property that GZ(G)≅Zp2⋊Zp2 is obtained. As a consequence, it is proved that such a group has exactly [(p+1)2+1] element centralizers and the structure of the commuting conjugacy class graph of G is completely determined.

The Wielandt subgroup of a finite group G is defined as w ( G ) = ∩ H ◁◁ G N G ( H ). In this paper, this subgroup is computed for certain finite groups.

Gyrogroups are new algebraic structures that appeared in 1988 in the study of Einstein’s velocity addition in the special relativity theory. These new algebraic structures were studied intensively by Abraham Ungar. The first gyrogroup that was considered into account is the unit ball of Euclidean space \mathbb{R}^3 ℝ3 endowed with Einstein’s velocity addition. The second geometric example of a gyrogroup is the complex unit disk \mathbb{D} 𝔻 ={z ∈ \mathbb{C}: |z|<1 ℂ:|z|<1 }. To construct a gyrogroup structure on \mathbb{D} 𝔻 , we choose two elements z_1, z_2 ∈\mathbb{D} z1,z2∈𝔻 and define the Möbius addition by z_1\oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2} z1⊕z2=z1+z21+z1‾z2 . Then (\mathbb{D},\oplus) (𝔻,⊕) is a gyrocommutative gyrogroup. If we define r \odot x r⊙x = = \frac{(1+|x|)^r - (1-|x|)^r}{(1+|x|)^r + (1-|x|)^r}\frac{x}{|x|} (1+|x|)r−(1−|x|)r(1+|x|)r+(1−|x|)rx|x| , where x ∈ \mathbb{D} x∈𝔻 and r ∈ \mathbb{R} r∈ℝ , then (\mathbb{D},\oplus,\odot) (𝔻,⊕,⊙) will be a real gyrovector space. This paper aims to survey the main properties of these Möbius gyrogroup and Möbius gyrovector space.

An irreducible character χ of a finite group G is called a Heisenberg character if Ker χ ⊇ [ G, [ G,G ]]. In this paper, we prove that the group G has exactly r , r ≤ 3, Heisenberg characters if and only if | G / G ′| = r . If G has exactly four Heisenberg characters, then | G / G ′| = 4, but the converse is not correct in general. Finally, it is proved that if G has exactly five Heisenberg characters, then | G / G ′| = 5 or | G / G ′| = 4 and one of the Heisenberg characters of G has the degree 2.

In this article we build a linear representation starting from a multigraph; this allows us to give an algebraic view of the multigraph we are studying. We show that two isomorphic multigraphs give equivalent representations; conversely two equivalent representations give isomorphic multigraphs. For the clarity of the article we give at the beginning, classical results on representations, nevertheless these are specific to our graph representation.

In 1869, Jordan proved that the set T of all finite groups that can be represented as the automorphism group of a tree is containing the trivial group, it is closed under taken the direct product of groups of lower orders in T , and wreath product of a member of T and the symmetric group on n symbols is again an element of T . The aim of this paper is to continue this work and another works by Klavik and Zeman in 2017 to present a class S of finite groups for which the automorphism group of each bicyclic graph is a member of S and this class is minimal with this property.

A group G is called a CA−group, if all the element centralizers of G are abelian and the commuting graph of G with respect to a subset A of G, denoted by Γ(G, A), is a simple undirected graph with vertex set A and two distinct vertices a and b are adjacent if and only if ab = ba. The aim of this paper is to generalize results of a recently published paper of F. Ali, M. Salman and S. Huang [On the commuting graph of dihedral group, Comm. Algebra 44 (6) (2016) 2389—2401] to the case that G is an CA−group.

Fora graph G of order n, size m and degree sequence D (G) = (d1,d2,...,dn), a new measure of irregularity IAG(G) = 1 - nn(d1 + r)(d2+ r) center dot center dot center dot (dn + r)/(2m + rn)n, r is an element of R?,0, is introduced. It is shown that if G has maximum IAG-irregularity among all con-nected graphs of order n and size m, then (i) Delta(G) = n - 1; (ii) for each u,v is an element of V(G) with the property dG(u) 5 dG(v), it holds that N(G,u) subset of N[G,v], where N(G,w) and N[G,w] are the neighbourhood and the closed neighbourhood of w in G, respectively; (iii) G is a threshold graph. Further, it is proven that if a graph H has a minimum value of IAG-irregularity among all irregular graphs of the same order and size, then Delta(H) - delta (H) = 1. Finally, the graphs with minimum and maximum IAG-irregularity in the classes of trees, unicyclic and bicyclic graphs are characterized.

Suppose that (T;*) is a groupoid with a left identity such that each element a 2 T has a left inverse. Then T is called a gyrogroup if and only if (i) there exists a function gyr : T x T -Aut(T) such that for all a; b; c 2 T, a * (b * c) = (a * b) ? gyr[a; b]c, where gyr[a; b]c = gyr(a; b)(c); and (ii) for all a; b 2 T, gyr[a; b] = gyr[a ? b; b]. In this paper, the structure of normal subgyrogroups of certain gyrogroups are investigated.

The MLS conjecture states that every finite simple group has a minimal logarithmic signature. The aim of this paper is proving the existence of a minimal logarithmic signature for some simple unitary groups PSUn(q). We report a gap in the proof of the main result of Hong et al. (Des. Codes Cryptogr. 77: 179–191, 2015) and present a new proof in some special cases of this result. As a consequence, the MLS conjecture is still open.

The commuting graph of a finite group $G$, $mathcal{C}(G)$, is a simple graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $xy = yx$. The aim of this paper is to compute the distance Laplacian spectrum and the distance Laplacian energy of the commuting graph of $CA$-groups.

The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation ⋆ containing an identity element such that each element has an inverse. Furthermore, for each pair (a,b) of elements of this structure there exists an automorphism a,b with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders p, 2p and p^2 are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.

A star coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] such that no path of length [Formula: see text] in [Formula: see text] is bicolored. In this paper, the star chromatic number of join of graphs is computed. Some sharp bounds for the star chromatic number of corona, lexicographic, deleted lexicographic and hierarchical product of graphs together with a conjecture on the star chromatic number of lexicographic product of graphs are also presented.

Let [Formula: see text] be a graph with edge set [Formula: see text]. For an edge [Formula: see text] in [Formula: see text], we define [Formula: see text], where [Formula: see text] and [Formula: see text] are degrees of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. For [Formula: see text], the graph invariants [Formula: see text], [Formula: see text] and [Formula: see text] are defined as [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and trees with the first through the third smallest [Formula: see text] and [Formula: see text], as well as the trees with the first through the forth smallest [Formula: see text] are also characterized.