# Ali Reza AshrafiUniversity of Kashan · Department of Pure Mathematics

Ali Reza Ashrafi

Professor of Mathematics

Editor-in-Chief of the Iranian Journal of Mathematical Chemistry

## About

502

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Introduction

I am working on Computational Group Theory, Mathematical Chemistry and Mathematical Physics.

## Publications

Publications (502)

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 var...

Let G be a finite simple graph with Laplacian polynomial ψ G , λ = ∑ k = 0 n − 1 n − k c k λ k . In an earlier paper, the coefficients c n − 4 and c n − 5 for forests with respect to some degree-based graph invariants were computed. The aim of this paper is to continue this work by giving an exact formula for the coefficient c n − 6 .

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 a...

The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\star$ 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 $\gyr{a,b}{}$ with this property that left associativity and left loop pr...

Suppose G is an undirected simple graph. A topological index for G is a number with this property that it is invariant under all graph isomorphisms with applications in chemistry. The Sombor and reduced Sombor indices of G are defined as S O G = ∑ u v ∈ E G d G 2 u + d G 2 v and S O red G = ∑ u v ∈ E G d G u − 1 2 + d G v − 1 2 , respectively. Here...

In this paper, we present exact formulae for p(G, (3), p(G, (4) and p(G, (5) in terms of some degree-based invariants, where G is an undirected simple graph, a k-subset of edges in G without common vertices is called a k-matching and the number of such subsets is denoted by p(G, k). Significance of research work Molecular descriptors play a signifi...

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 in...

A graph G is determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomor-phic to G. A multi-fan graph is a graph of the form (P n1 ∪ P n2 ∪ · · · ∪ P n k) ▽ K 1 , where K 1 denotes the complete graph of size 1, P n1 ∪ P n2 ∪ · · · ∪ P n k is the disjoint union of paths P ni , n i ≥ 1 and 1 ≤ i ≤ k, and a sta...

The number of subgroups, normal subgroups and characteristic subgroups of a finite group G are denoted by Sub(G), N Sub(G) and CSub(G), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences U 6n , V 8n and H(n) of groups that can be...

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the...

In this article, we study the resolvent energy and the pseudospectrum energy of a sequence C12n of fullerenes with exactly 12n carbon atoms. In particular, the resolvent energy of these fullerenes together with their lower bounds are obtained.

Let G be a graph with vertex set V (G) and edge set E(G). The vertex-edge degree of the vertex v, d e G (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, d v G (e), equals to the number of vertices of the union of the open neighborhood of u...

Let G be a graph with vertex set V (G) and edge set E(G). The vertex-edge degree of the vertex v, d e G (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, d v G (e), equals to the number of vertices of the union of the open neighborhood of u...

Suppose $G$ is a undirected simple graph. A $k-$subset of edges in $G$ without common vertices is called a $k-$matching and the number of such subsets is denoted by $p(G,k)$. The aim of this paper is to present exact formulas for $p(G,3)$, $p(G,4)$ and $P(G,5)$ in terms of some degree-based invariants.

Suppose G is a simple graph with edge set EG. The Randić index RG is defined as RG=∑uv∈EG1/degGudegGv, where degGu and degGv denote the vertex degrees of u and v in G, respectively. In this paper, the first and second maximum of Randić index among all n−vertex c−cyclic graphs was computed. As a consequence, it is proved that the Randić index attain...

Suppose that $(T,\star)$ is a groupoid with a left identity such that each element $a\in T$ has a left inverse. Then $T$ is called a \textit{gyrogroup} if and only if $(i)$ there exists a function $gyr:T\times T\longrightarrow Aut(T)$ such that for all $a,b,c\in T$, $a\star(b\star c)= (a\star b)\star gyr[a,b]c$, where $gyr[a,b]c=gyr(a,b)(c)$; and $...

A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to G. In some recent papers it is proved that the friendship graphs and starlike trees are DLS. If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than two, then the resul...

In this paper, the resolvent energy and spectral moment are investigated by the help of some special functions. Some sharp bounds are analyzed for these structures including its vertices, its edges, its degrees and its eigenvalues.

Let $G$ be a finite simple graph with Laplacian polynomial $\psi(G,\lambda)=\sum_{k=0}^n(-1)^{n-k}c_k\lambda^k$. In an earlier paper, the coefficients $c_{n-4}$ and $c_{n-5}$ for tree with respect to some degree-based graph invariants were computed. The aim of this paper is to continue this work by giving an exact formula for the coefficients $c_{n...

Let G be a graph with edge set EG and e=uv∈EG. Define nue,G and mue,G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v, respectively. The numbers nve,G and mve,G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as MoG=∑uv∈EGnuuv,G−n...

In 1869, Jordan proved that the set $\mathcal{T}$ of all finite group that can be represented as the automorphism group of a tree is containing the trivial group and it is closed under taken direct product of groups of lower order in $\mathcal{T}$ and wreath product of a member in $\mathcal{T}$ and the symmetric group on $n$ symbols. The aim of thi...

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 som...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The Sombor and reduced Sombor indices of $G$ are defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{deg_G(u)^2+deg_G(v)^2}$ and $SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(deg_G(u)-1)^2+(deg_G(v)-1)^2}$, respectively. We denote by $H_{n,\nu}$ the graph constructed from the star $S_n$ by adding $\nu$ edg...

In this paper, a 2-gyrogroup G(n) of order 2n, n≥3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use...

Let G be a finite group and \(N \unlhd G\). The normal subgroup based power graph of G, \(\Gamma _N(G)\), is an undirected graph with vertex set \((G{\setminus }N) \cup \{e\}\) in which two distinct vertices a and b are adjacent if and only if \(aN = b^mN\) or \(bN = a^nN\), for some positive integers m and n. The aim of this paper is to characteri...

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 $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a grou...

Suppose that $G$ is a groupoid with binary operation $\otimes$. The pair $(G,\otimes)$ is said to be a gyrogroup if the operation $\otimes$ has a left identity, each element $a \in G$ has a left inverse and the gyroassociative law and the left loop property are satisfied in $G$. In this paper, a method for constructing new gyrogroups from old ones...

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

Combinatorial techniques based on Sheehan’s modification of Pόlya’s theorem and Mȍbius inversion technique together with character cycle indices are applied to face colorings of giant fullerenes. These techniques are applied to icosahedral fullerenes, C80 with a chamfered dodecahedron structure, a chiral fullerene C140, icosahedral C180 and C240 wi...

Suppose G is a simple graph, \(\Gamma \le Aut(G)\) and \(\alpha \in \Gamma \). Define \(\mu (G)=\sum _{u \in V(G), \alpha \in \Gamma } d(u,\alpha (u))\), \(\eta (G)=\sum _{u \in V(G), \alpha \in \Gamma } ( d(u,\alpha (u)))^2\), \(GP(G)= \frac{|V(G)|}{2|\Gamma |}\mu (G)\) and \(GP^{(2)}(G) = \frac{1}{2}GP(G) + \frac{|V(G)|}{4|\Gamma |}\eta (G)\). Th...

Suppose G is a finite group. The set of all centralizers of 2-element subsets of G is denoted by 2−Cent(G). A group G is called (2,n)‐centralizer if |2−Cent(G)|=n and primitive (2,n)‐centralizer if |2−Cent(G)|=|2−Cent(GZ(G))|=n, where Z(G) denotes the center of G. The aim of this paper is to present the main properties of (2,n)‐centralizer groups a...

The workshop focuses on the latest advanced topics in Dynamical Systems Theory, Group Theory and Functional Analysis . This event brings together professors, researchers, scholars, and students in the mentioned fields to share experience, foster collaborations across industry and academia. The primary aim of organizing such an event is to continue...

Let G be a nite group and C(G) be a family of representatives of the conjugacy classes of subgroups in G.

Let G be a chemical graph with vertex set {v1,v1,…,vn} and degree sequence d(G)=(degG(v1),degG(v2),…,degG(vn)).
The inverse degree, R(G) of G is defined as
R(G)=∑ni=11degG(vi). The cyclomatic number of G is defined as γ=m−n+k, where m, n and k are the number of edges, vertices and components of G, respectively. In this paper, some upper
bounds on t...

Неприводимый характер $\chi$ конечной группы $G$ называется характером Гейзенберга, если $\ker \chi \supseteq [G, [G, G]]$. В статье доказано, что группа $G$ имеет в точности $r$, $r \leq 3$, характеров Гейзенберга тогда и только тогда, когда $|{G}/{G'}|=r$. Если $G$ имеет в точности четыре характера Гейзенберга, то $|{G}/{G'}|=4$, но обратное в об...

Hyper-adamantane is an adamantine in which vertices are changed by a cell/shape. Some mathematical properties of hyper-adamantanes built with several symmetrical shapes are detailed.

In 2008, Diaconis and Isaacs introduced the notion of a supercharacter theory of a finite group in which supercharacters replace with irreducible characters and superclasses by conjugacy classes. In this paper, we introduce an algorithm for constructing supercharacter theories of a finite group by which all supercharacter theories of groups contain...

Let G be a graph with vertex set V(G). The total irregularity of G is defined as \(irr_t(G)=\sum _{\{u,v\}\subseteq V(G)}|deg_G(u)-deg_G(v)|\), where \(deg_G(v)\) is the degree of the vertex v of G. The cyclomatic number of G is defined as \(c = m - n + k\), where m, n and k are the number of edges, vertices and components of G, respectively. In th...

Suppose $G$ is a finite group. The set of all centralizers of $2-$element subsets of $G$ is denoted by $2-Cent(G)$. A group $G$ is called $(2,n)-$centralizer if $|2-Cent(G)| = n$ and primitive $(2,n)-$centralizer if $|2-Cent(G)| = |2-Cent(\frac{G}{Z(G)})| = n$, where $Z(G)$ denotes the center of $G$. The aim of this paper is to present the main pro...

It is our pleasure to welcome you in the "2nd International Workshop on Advanced Topics in Dynamical Systems, IWATDS", March 1-2, 2020.
The 2nd IWATDS 2020 is a sequel to the Workshop on Advanced Topics in Dynamical Systems Theory 2019 which was organized on third of March 2019 by the Department of Mathematics, Faculty of Computer Science and Math...

Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the Conjecture 4.2 of [J A de Oliveira, C S Oliveira, C Justel and N M Maia de Abreu, Measures of irregularity of graphs...

A prose graph Γ = RG(a 3 , a 4 ,. .. , a s) is a graph consisting of p = a 3 + a 4 + · · · + a s ≥ 2 cycles that all meet in one vertex, and a i (3 ≤ i ≤ s) is the number of cycles in Γ of length i. A graph G is said to be DLS (resp. DQS) if it is determined by the spectrum of its Laplacian (resp. signless Laplacian) matrix, i.e., if every graph wi...

A group $H$ is said to be capable, if there exists another group$G$ such that $\frac{G}{Z(G)}~\cong~H$, where $Z(G)$ denotes thecenter of $G$. In a recent paper \cite{2}, the authorsconsidered the problem of capability of five non-abelian $p-$groups of order $p^4$ into account. In this paper, we continue this paper by considering three other groups...

In 2008, Diaconis annd Isaacs introduced the notion of a supercharacter theory of a finite group in which supercharacters replace with irreducible characters and superclasses by conjugacy classes. In this paper, we introduce an algorithm for constructing supercharacter theories of a finite group by which all supercharacter theories of groups contai...

Let G be a graph with vertex set V(G). The total irregularity of G is defined as irrt(G)=∑{u,v}⊆V(G)|degG(u)−degG(v)|, where degG(v) is the degree of the vertex v of G. The aim of this paper is to present some bounds for this graph invariant. A new simple proof for a recently proposed conjecture on total irregularity of graphs is also presented.

A simple connected graph G is called a p-quasi k-cyclic graph, if there exists a subset S of vertices such that |S|=p, G-S is k-cyclic and there is no a subset S` of V(G) such that |S`|<|S| and G-S` is k-cyclic. The aim of this paper is to characterize graph with maximum values of Zagreb indices among all p-quasi k-cyclic graph of order k>=3.

If q copies of K 1,p and a cycle C q are joined by merging any vertex of C q to the vertex with maximum degree of K 1,p , then the resulting graph is called the jellyfish graph JF G(p, q) with parameters p and q. Two graphs are said to be Q-cospectral (respectively, L-cospectral) if they have the same signless Laplacian (respectively, Laplacian) sp...

In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS and DQS.

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 $PSU_{n}(q)$. We report a gap in the proof of the main result of [H. Hong, L. Wang, Y. Yang, Minimal logarithmic signatures for the unitary g...

Suppose G is a molecular graph with edge set E(G). The hyper-Zagreb index of G is defined as HM (G) = uv∈E(G) [degG(u) + degG(v)] 2 , where degG(u) is the degree of a vertex u in G. In this paper, all chemical trees of order n ≥ 12 with the first twenty smallest hyper-Zagreb index are characterized.

Suppose $G$ is a simple graph with edge set $E(G)$. The Randi\'{c} index $R(G)$ is defined as $R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{deg_{G}(u)deg_{G}(v)}}$, where $deg_G(u)$ denotes the vertex degree of $u$ in $G$. In this paper, the first and second maximum of Randi\'{c} index among all $n-$vertex $k-$cyclic graphs were computed.

Suppose G is a nite group and max(G), nmax(G), snmax(G), maxn(G) and minn(G) are denoted the number of maximal, normal maximal, self-normalizing maximal, maximal normal and minimal normal subgroups of G, respectively. The aim of this paper is to compute these numbers for certain classes of nite groups.

The supercharacter theory of a finite group is a generalization of the ordinary character theory of finite groups that was introduced by Diaconis and Isaacs in 2008. In this paper, the concept of groups with quasi-identical character tables are presented. It is proved that the groups with quasi-identical character tables have the same number of sup...

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d G (u, v) = 3}, where d G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polari...

The Szeged index Sz(G) of a simple connected graph G is the sum of the terms nu(e)nv (e) over all edges e = uv of G, where nu(e) is the number of vertices of G lying closer to u than v, and nv (e) is defined analogously. The aim of this paper is to present some relationship between Szeged index and some of its variants such as the edge-vertex Szege...

A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is known to be the case only for a very few families. In some recent papers it is proved that the friendship graphs...

Suppose R is a ring. The multiplicative power graph P(R) of R is the graphwhose vertices are elements of R, where two distinct vertices x and y are adjacent if and only if there exists a positive integer n such that x^n = y or y^n = x. In this paper, the tensor product of the power graphs of some nite rings and also some main properties of them wil...

The irregularity of a graph $G$ is defined as $irr(G)=\sum_{uv\in E(G)}|d_u- d_v|$, where $d_u$ denotes the degree of a vertex $u\in V(G)$ and $E(G)$ is the edge set of $G$. From the class of all $n$-vertex (molecular) trees, graphs with the first five minimal $irr$-values have already been characterized in the literature. The main purpose of this...

Abstract. Let G be an n-vertex graph with the vertex degree sequence d1, d2,… dn. The Narumi{Katayama index of G is defined
as NK(G) =∏_(i=1)^n▒dn. We determine eight classes of n-vertex tricyclic
graphs with the first through the eighth maximal NK index, n ≥10.
We also identify ten classes of n-vertex tetracyclic graphs with the first
through the...

Suppose G is a simple graph with vertex set and edge set V (G) and E(G), respectively. Define B(G) to be the set of all {x, y} such that {x, y} ⊆ V (G) ∪E(G) and members of {x, y} are adjacent or incident to each other. Alwardi et al. in a recent paper, [A. Alwardi, A. Alqesmah, R. Rangarajan and I. N. Cangul, Entire Zagreb indices of graphs, Discr...

In this paper, some inequalities between the Wiener, hyper-Wiener, first Zagreb, second Zagreb, first reformulated Zagreb, second reformulated Zagreb and the general Zagreb indices of a simple graph are given. Our results improve some earlier bonds between these graph invariants.

The Graovac-Pisanski index is a symmetry-based version of the well-known Wiener topological index. The aim of this paper is to compute this invariant for both armchair and zig-zag polyhex carbon nanotubes marking important theoretical considerations on their relative stability and the effects of the edge states. We also compare the Graovac-Pisanski...

A (k,6)-fullerene graph is a planar 3-connected cubic graph whose faces are k-gons and hexagons. The aim of this paper is to compute the equitable chromatic, b-chromatic and list chromatic numbers of some fullerenes.

Let G be a finite group. The commuting graph Γ=C(G) is a simple graph with vertex set G and two vertices are adjacent if and only if they commute with each other. In this paper, the normalized and signless Laplacian spectra of the commuting graphs of certain classes of finite groups are computed.

Let G be a graph with edge set E(G). The Randić and sum-connectivity indices of G are defined as R(G)=∑uv∈E(G)[Formula presented] and SCI(G)=∑uv∈E(G)[Formula presented], respectively, where degG(u) denotes the vertex degree of u in G. In this paper, the extremal Randić and sum-connectivity index among all n-vertex chemical trees, n ≥ 13, connected...

Let G be a finite group and c(G) denote the number of cyclic subgroups of G. The group G is called an n-cyclic group if \(c(G) = n\). In an earlier paper, finite n-cyclic groups with \(n \le 8\) are classified and a characterization of the alternating group on five symbols based on the number of cyclic subgroups is given. The aim of this article is...

Let G be a simple and undirected graph with Laplacian polynomial . In earlier works, some formulas for computing , and in terms of the number of vertices, the Wiener, the first Zagreb and the forgotten indices are given. In this paper, we continue this work by computing , where T is a tree. A lower and an upper bound for are obtained.

The first degree-based entropy of a connected graph G is defined as: \(I_1(G)=\log (\sum _{v_i\in V(G)}\deg (v_i))-\sum _{v_j\in V(G)}\frac{\deg (v_j)\log \deg (v_j)}{\sum _{v_i\in V(G)}\deg ( v_i)}\). In this paper, we apply majorization technique to extend some known results about the maximum and minimum values of the first degree-based entropy o...

In a seminal paper published in 1998, Shinsaku Fujita introduced the concept of Q-conjugacy character table of a finite group. He applied this notion to solve some problems in combinatorial chemistry. In this paper, the Q-conjugacy character table of dihedral groups is computed in general. As a consequence, Q-conjugacy character table of molecules...

The aim of this paper is to present some inequalities between degree distance and Gutman index with the Zagreb and reformulated Zagreb indices of graphs.

Suppose G is a graph with vertex set V(G). The Graovac-Pisanski index of G is defined as GP(G) = 1/2|V(G)|²δ(G), where (Eqution pressented) This is a type of graph invariant that is combined distance and symmetry of molecules under consideration. The aim of this paper is to compute the symmetry groups and Graovac-Pisanski index of some linear polym...

The Graovac�Pisanski index is a type of graph invariant that is combined distance and symmetry of molecules under consideration. The aim of this paper is to compute the symmetry groups and Graovac-Pisanski index of some linear polymers.

Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A ∪ B ≤ N. The aim of this paper is to study the normal graph of a finite...

The deck of a graph $X$, $D(X)$, is defined as the multiset of all vertex-deleted subgraphs of $X$. Two graphs are said to be hypomorphic, if they have the same deck. Kelly-Ulam conjecture states that any two hypomorphic graphs on at least three vertices are isomorphic. In this paper, we first prove that for two finite simple hypomorphic graphs the...

Suppose G is a finite group with identity element 1. The generating graph \(\Gamma (G)\) is defined as a graph with vertex set G in such a way that two distinct vertices are connected by an edge if and only if they generate G and the Q-generating graph \(\Omega (G)\) is defined as the quotient graph \(\frac{\Gamma (G)\backslash \{ 1\}}{\mathcal {C}...

Let G be a finite group and �(G; S) is the Ggraph of a group G with respect
to a non-empty subset S. The aim of this paper is to study the structure and the
automorphism group of a simple form of Ggraph for some finite groups like
alternating group, dihedral, semi-dihedral, dicyclic, Zmo�Z2n, where � is inverse
mapping and V8n = fa; bja2n = b4 =...

For a molecular graph G with vertex set V(G), the Wiener polarity index Wp(G) is the number of unordered pairs of vertices {u, v} such that . In this paper, an ordering of chemical trees of order n with respect to Wiener polarity index is given.

Suppose $X$ is a simple graph. The $X-$join $\Gamma$ of a set of complete or empty graphs $\{X_x \}_{x \in V(X)}$ is a simple graph with the following vertex and edge sets: \begin{eqnarray*} V(\Gamma) &=& \{(x,y) \ | \ x \in V(X) \ \& \ y \in V(X_x) \},\\ E(\Gamma) &=& \{(x,y)(x^\prime,y^\prime) \ | \ xx^\prime \in E(X) \ or \ else \ x = x^\prime \...

All graphs considered are assumed to be simple and finite. The sets of vertices and edges of a graph G are denoted by () and (), respectively. By and we denote the number of vertices and edges of, ie,=|()| and=|()|. If has components, then=()=−+ is called the cyclomatic number of. In this work we shall be mainly concerned with connected graphs, for...

A graph is said to be a semi-Cayley graph over a group G if it admits G as a semiregular automorphism group with two orbits of equal size. In this paper, definition of normal edge-transitive, arc-transitive and (Formula presented.)-arc-transitive semi-Cayley graphs are given. The main properties of such semi-Cayley graphs are shown and a condition...

The power graph $\mathcal{P}(G)$ is a graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is a graph with vertex set $G$ in which two elements $x, y \in G$ are joined if $o(x) | o(y)$ or $o(y) | o(x)$. The purpose of this pape...

A topological index is called vertex-degree-based if it can be deffined by vertex degrees. The harmonic, atom-bond connectivity and Randic indices are three important examples of such topological indices. The aim of this paper is to ffind lower and upper bounds for Randic, harmonic and atom-bond connectivity indices of edge corona product of graphs...

The variable sum exdeg index of a graph G is defined as where a ≠ 1 is a positive real number. The aim of this paper is applying the majorization technique to obtain the maximum and minimum values of variable sum exdeg index of trees, unicyclic, bicyclic and tricyclic graphs.

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and the greatest common divisor of $a$ and $b$, respectively. The aim of this paper is to first present a new pro...

The power graph $\mathcal{P}(G)$ of a group $G$ is the graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The aim of this paper is to compute the automorphism group of the power graph of several well-known and important classes of finite groups.

For a finite group , the power graph is a graph with the vertex set , in which two distinct elements are adjacent if one is a power of the other. Feng, Ma and Wang (Feng et al., 2016) described the full automorphism group of . In this paper, we study automorphism groups of the main supergraphs and cyclic graphs, which are supergraphs of . It is pro...

Let G be a graph and k is a positive integer. Hammack and Livesay in [The inner power of a graph, Ars Math. Contemp., 3 (2010), no. 2, 193-199] introduced a new graph operation G^{(k)}, called the k^{th} inner power of G. In this paper, it is proved that if G is bipartite then G^{(2)} has exactly three components such that one of them is bipartite...

The purpose of this paper is computing the fundamental relations and automorphism groups of very thin Hv-groups. In this regards, we first investigate some basic properties of Hv-groups and then we show that any given group is isomorphic to the fundamental group of a nontrivial Hv-group. The main properties of very thin Hv-groups are also investiga...

The aim of this paper is to first establish some new bounds for the resolvent Estrada index of graphs. Finally, we introduce the resolvent signless Laplacian Estrada index of a graph and the extremal trees together with some bounds in general with respect to this new invariant are presented. © 2017, University of Kragujevac, Faculty of Science. All...

In chemical graph theory, two different sets of objects are usually considered: atoms and chemical bonds. Define V and E as the set of all atoms and chemical bonds. Then the pair G = (V,E) has a topological structure named “molecular graph”. A chemical bond e connects two atoms u and v and it is convenient to write e = uv. We also use the terms ver...

The aim of this paper is to present a general method which is useful for computing the symmetry group of a molecule constructed from cycloalkynes by substituting alkyl instead of hydrogen atoms.