Index of Graphs

Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(n eu (e|G)+ n ev (e|G)), where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we first compute the PI index of a class of pericondensed benzenoid graphs consisting of n rows, n ≤ 3, of hexagons of various lengths. Finally, we prove that for any connected graph G with exactly m edges, PI(G) ≤ m(m-1) with equality if and only if G is an acyclic graph or a cycle of odd length. 1 . Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index 2-17 . This newly proposed topological index, differ from the Wiener index 18 , the oldest topological index for acyclic (tree) molecules. We now describe some notations which will be adhered to throughout. Benzenoid systems (graph representations of benzenoid hydrocarbons) are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin 19 and in the references cited therein. Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and nev(e|G). neu(e|G) is the number of edges lying closer to vertex u than to vertex v, and nev(e|G) is the number of edges lying closer to vertex v than to vertex u. Then PI(G) = ∑(neu(e|G) + nev(e|G)), where the summation goes over all edges of G. Edges equidistant from both ends of the edge e = uv are not counted and the number of such edges is denoted by N(e). To clarify this, for every vertex u and any edge f = zw of graph G, we define d(f,u) = Min{d(u,w),d(u,z)}. Then f is equidistant from both ends of the edge e = uv if d(f,u) = d(f,v). In a series of papers, Khadikar and coauthors 2-17 defined and then computed the PI index of some chemical graphs. The present author 20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature. 21,22

In this paper, we have proposed new windmill graph, that is Basava wheel windmill graph. The Basava wheel windmill graph \(W^{(m)}_{n+1}\) is the graph obtained by taking \(m\geq 2\) copies of the graph \(K_1+W_{n}\) for \(n\geq 4\) with a vertex \(K_1\) in common. Inspired by recent work on topological indices, proposed new degree-based topological indices namely, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of a graph \(G\). We have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randi\(\acute{c}\) index, general Randi\(\acute{c}\) index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, Symmetric division deg index, Sombor index, SK indices, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of Basava wheel windmill graph. Further, we have computed exact values of these topological indices of chloroquine, hydroxychloroquine and remdesiver.

The vertex PI index and Szeged index of bridge graphs

In this paper, the relation between the reciprocal Randic index, Reduced reciprocal Randic index and Atom-bond connectivity index of a simple connected graph and its thorn graph is stablished and the atom-bond connectivity \((A B C)\) index of a graph \(G\) is defined as \(A B C(G)=\sum_{w v \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_u d_v}}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\) [13]. Reciprocal Randic \((R R)\) index of a graph \(G\) is defined as \(R R(G)=\sum_{w v \in E(G)} \sqrt{d_u d_v}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\). Reduced Reciprocal Randic \((R R R)\) index of a graph \(G\) is defined as \(\operatorname{RRR}(G)=\sum_{w v \in E(G)} \sqrt{\left(d_u-1\right)\left(d_v-1\right)}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\). Results are applied to compute the reciprocal Randic index, Reduced reciprocal Randic index and Atom-bond connectivity index of thorn rings, thorn paths, thorn rods, thorn star, thorn star \(S_n\left(p_1, p_2, \cdots, p_{n-1}, p_n\right)\).

A lower bound on the modified Randić index of line graphs

The concept of the topological index of a graph is increasingly diverse because researchers continue to introduce new concepts of topological indices. Researches on the topological indices of a graph which initially only examines graphs related to chemical structures begin to examine graphs in general. On the other hand, the concept of graphs obtained from an algebraic structure is also increasingly being introduced. Thus, studying the topological indices of a graph obtained from an algebraic structure such as a group is very interesting to do. One concept of graph obtained from a group is subgroup graph introduced by Anderson et al in 2012 and there is no research on the topology index of the subgroup graph of the symmetric group until now. This article examines several topological indices of the subgroup graphs of the symmetric group for trivial normal subgroups. This article focuses on determining the formulae of various Zagreb indices such as first and second Zagreb indices and co-indices, reduced second Zagreb index and first and second multiplicatively Zagreb indices and several eccentricity-based topological indices such as first and second Zagreb eccentricity indices, eccentric connectivity, connective eccentricity, eccentric distance sum and adjacent eccentric distance sum indices of these graphs.

Vertex and edge PI indices of Cartesian product graphs

Inverse sum indeg status index of graphs and its applications to octane isomers and benzenoid hydrocarbons

Equitable neighbour-sum-distinguishing edge and total colourings

Let G be a molecular graph. The distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. In this paper, we consider some infinite families of 3-fence graphs namely ladder, circular ladder and Mobius ladders. We compute the eccentricity based topological indices of these graphs and their line graphs. Also, we study the relation between the indices of these graphs with their line graphs. Furthermore, we construct a square grid from the ladder graph and study the eccentricity based topological indices for this grid graph and its line graph.

<abstract><p>The bond incident degree (BID) index of a graph $ G $ is defined as $ BID_{f}(G) = \sum_{uv\in E(G)}f(d(u), d(v)) $, where $ d(u) $ is the degree of a vertex $ u $ and $ f $ is a non-negative real valued symmetric function of two variables. A graph is stepwise irregular if the degrees of any two of its adjacent vertices differ by exactly one. In this paper, we give a sharp upper bound on the maximum degree of stepwise irregular graphs of order $ n $ when $ n\equiv 2({\rm{mod}}\;4) $, and we give upper bounds on $ BID_{f} $ index in terms of the order $ n $ and the maximum degree $ \Delta $. Moreover, we completely characterize the extremal stepwise irregular graphs of order $ n $ with respect to $ BID_{f} $.</p></abstract>

For a simple connected graph G, center C(G) and periphery P(G) are subgraphs induced on vertices of G with minimum and maximum eccentricity, respectively. An n-vertex graph G is said to be an almost self-centered (ASC) graph if it contains \(n-2\) central vertices and two peripheral (diametral) vertices. An ASC graph with radius r is known as an r-ASC graph. The r-ASC index of any graph G is defined as the minimum number of new vertices, and required edges, to be introduced to G such that the resulting graph is r-ASC graph in which G is induced. For \(r=2,3\), r-ASC index of few graphs is calculated by Klavžar et al. (Acta Mathematica Sinica, 27:2343–2350, 2011 [1]), Xu et al. (J Comb Optim 36(4):1388–1410, 2017 [2]), respectively. Here we give bounds to r-ASC index of diameter two graphs and determine the exact value of this index for paths and cycles.

Some results on the index of unicyclic graphs