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‎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 an $n-$vertex graph with $m$ vertices‎. ‎The degree deviation measure of $G$ is defined as‎ ‎$s(G)$ $=$ $\sum_{v\in V(G)}|deg_G(v)‎- ‎\frac{2m}{n}|,$ 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, Pesq‎. ‎Oper.‎, ‎33 (2013) 383--398]‎. ‎The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed‎.

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 resulting graph is called a path-friendship graph. In this paper, it is proved that the path-friendship graphs, a natural generalization of friendship graphs and starlike trees, are also DLS. Consequently, using these results we provide a solution for an open problem.

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 SOG = ∑uv ∈EGdG2u+dG2v and SOredG=∑uv ∈EGdGu−12+dGv−12 , respectively. Here, dGu denotes the degree of the vertex u in G. In this paper, these invariants were computed for alkanes, alkyls, and annulenes. A comparison of our calculations with molecular mass is also presented. As a consequence, it is shown that there is a good correlation between Sombor index and molecular mass of these compounds.

Let G be a finite simple graph with Laplacian polynomial ψG,λ=∑k=0n−1n−kckλk. In an earlier paper, the coefficients cn−4 and cn−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 cn−6.

‎Let $G$ be a simple graph‎. ‎The graph $G$ is called a quasi unicyclic graph if there exists a vertex $x \in V(G)$ such that $G-x$ is a connected graph with a unique cycle‎. ‎Moreover‎, ‎the first and the second Zagreb indices of $G$ denoted by $M_1(G)$ and $M_2(G)$‎, ‎are the sum of $\deg^2(u)$ overall vertices $u$ in $G$ and the sum of $\deg(u)\deg(v)$ of all edges $uv$ of $G$‎, ‎respectively‎. ‎The first and the second Zagreb indices are defined relative to the degree of vertices‎. ‎In this paper‎, ‎sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given‎.

An irreducible character χ of a group G is called a Heisenberg character, if Kerχ ⊇ [G,[G,G]]. In this paper, the Heisenberg characters of the quaternion Heisenberg, generalized Heisenberg, polarised Heisenberg and three other types of infinite Heisenberg groups are computed.

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−nvuv,G and MoeG=∑uv∈EGmuuv,G−mvuv,G, respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.

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 degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

The power graph $\mathcal{P}(G)$ of a group $G$ is the graphwith group elements as vertex set and two elements areadjacent 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.

‎The first Zagreb index‎, ‎$M_1(G)$‎, ‎and second Zagreb index‎, ‎$M_2(G)$‎, ‎of the graph $G$ is defined as $M_{1}(G)=sum_{vin‎ ‎V(G)}d^{2}(v)$ and $M_{2}(G)=sum_{e=uvin E(G)}d(u)d(v),$ where‎ ‎$d(u)$ denotes the degree of vertex $u$‎. ‎In this paper‎, ‎the first‎ ‎and second maximum values of the first and second Zagreb indices‎ ‎in the class of all $n-$vertex tetracyclic graphs are presented‎.

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 the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups.

Darafsheh and Assari in [Normal edge-transitive Cayley graphs onnon-abelian groups of order 4p, where p is a prime number,Sci. China Math. {bf 56} (1) (2013) 213$-$219.] classified the connected normal edge transitive and$frac{1}{2}-$arc-transitive Cayley graph of groups of order$4p$. In this paper we continue this work by classifying theconnected Cayley graph of groups of order $2pq$, $p > q$ areprimes. As a consequence it is proved that $Cay(G,S)$ is a$frac{1}{2}-$edge-transitive Cayley graph of order $2pq$, $p> q$ if and only if $|S|$ is an even integer greater than 2, $S =T cup T^{-1}$ and $T subseteq { cba^{i} | 0 leq i leq p- 1}$ such that $T$ and $T^{-1}$ are orbits of $Aut(G,S)$ andbegin{eqnarray*}G &=& langle a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r rangle,G &=& langle a, b, c | a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r rangle,end{eqnarray*}where $r^q equiv 1 (mod p)$.

Suppose $Gamma$ is a graph with $V(Gamma) = { 1,2, cdots, p}$and $ mathcal{F} = {Gamma_1,cdots, Gamma_p} $ is a family ofgraphs such that $n_j = |V(Gamma_j)|$, $1 leq j leq p$. Define$Lambda = Gamma[Gamma_1,cdots, Gamma_p]$ to be a graph withvertex set $ V(Lambda)=bigcup_{j=1}^pV(Gamma_j)$ and edge set$E(Lambda)=big(bigcup_{j=1}^pE(Gamma_j)big)cupbig(bigcup_{ijinE(Gamma)}{uv;uin V(Gamma_i),vin V(Gamma_j)}big) $. Thegraph $ Lambda$ is called the $Gamma-$join of $ mathcal{F}$.The power graph $mathcal{P}(G)$ of a group $G$ is the graphwhich has the group elements as vertex set and two elements areadjacent if one is a power of the other. The aim of this paper isto prove $mathcal{P}(mathbb{Z}_{n}) = K_{phi(n)+1} +Delta_n[K_{phi(d_1)},K_{phi(d_2)},cdots, K_{phi(d_{p})}]$,where $Delta_n$ is a graph with vertex and edge sets $V(Delta_n)={d_i | 1,nnot = d_i | n, 1leq ileq p}$ and $ E(Delta_n)={ d_id_j | d_i|d_j, 1leq i

Let G be an n-vertex graph with degree sequence d1, d2, …, dn. The harmonic index H(G) is defined as n/I(G)= Σ n i= 1(l/di)., In this paper the top ten values of harmonic index in the set of all chemical trees of order n are determined.

The corona product $Gcirc H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$; and by joining each vertex of the $i$-th copy of $H$ to the $i$-th vertex of $G$, where $1 leq i leq |V(G)|$. In this paper, exact formulas for the eccentric distance sum and the edge revised Szeged indices of the corona product of graphs are presented. We also study the conditions under which the corona product of graphs produces a median graph.

Bounds for the degree Kirchhoff index of the line and para-line graphs are determined.The special case of regular graphs is analyzed in due detail.

The eccentric connectivity index ξ(G) of the graph G is defined as ξ(G) = Σu∈V(G) deg(u)ε(u) where deg(u) denotes the degree of vertex u and ε(u) is the largest distance between u and any other vertex v of G. In this paper an exact expression for the eccentric connectivity index of an armchair polyhex nanotube is given.

Hetero-fullerenes are fullerenes in which some of the carbon atoms are replaced by other atoms. This paper uses the Pólya theorem to count the number of their possible positional isomers and chiral isomers. To do this, the computer algebra system GAP was applied to compute this number for a class of IPR hetero-fullerenes with Ih point group symmetry. These fullerenes were constructed by means of the leapfrog principle.

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2.

The Wiener and detour indices of a molecular graph G are defined as the sum of the lengths of all shortest and longest paths between the vertices of G. In this paper the exact formulae for the Wiener and detour indices of a new type of nanostar dendrimers are given.

An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix. Balasubramanian (1995) computed the Euclidean graphs and their automorphism groups for benzene, eclipsed and staggered forms of ethane, and eclipsed and staggered forms of ferrocene. It was shown by Balasubramanian that the molecular symmetry groups can be obtained as the automorphism groups of edge-weighted Euclidean graphs. In this paper we calculate the atom centers of hexamethylcyclohexane molecule using the chemistry package HyperChem and then compute its symmetry group.

Graph theory provides an elegant and natural representation of molecular symmetry and the resulting group expressed in terms of permutations is isomorphic to the permutation-inversion group of Longuet-Higgins. In this paper, using the group theory package GAP, we compute the automorphism group of the Euclidean graph of benzenoid chains.