On the degree-Kirchhoff index, Gutman index and the Schultz index of pentagonal cylinder/ M\"{o}bius chain

The degree-Kirchhoff index of a graph is given by the sum of inverses of non-zero eigenvalues of the normalized Laplacian matrix of the graph multiplied with the total degree of the graph. Explicit formulas for the degree-Kirchhoff index of various types of cylinder chain and M\"{o}bius chain have been obtained by many researchers in the recent past. In the present paper, we obtain closed-form formulas for the degree-Kirchhoff index of pentagonal cylinder/ M\"{o}bius chain. Also we find here the Gutman index and the Schultz index for those graphs

New bounds of extended energy of a graph

Extended adjacency matrix of a graph with $n$ vertices is a real symmetric marix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when $i,j$ are adjacent, and zero otherwise. Aggregate of absolute eigenvalues of the extended adjacency matrix is termed as the extended energy. In this paper, the concept of extended vertex energy is introduced and some bounds of extended vertex energy are obtained. From there, we obtain some new upper bounds of the extended energy of a graph. Next, we obtain two inequalities which relate the extended energy with the ordinary graph energy. One of those inequalities resolves a conjecture which states that for every graph, ordinary energy can never exceed the extended energy. Using the relationships of extended energy and ordinary energy, we obtain new bounds of extended energy involving the order, size, largest and smallest degree of the graph. We show that these new bounds are improvements of some existing bounds. Finally, some improved bounds of Nordhaus-Gaddum-type are also found.Comment: 17 page

Computing the F-index of nanostar dendrimers

AbstractDendrimers are highly branched nanostructures and are considered a building block in nanotechnology with a variety of suitable applications. In this paper, a vertex degree-based topological index, namely, the F-index, which is defined as the sum of cubes of a graph's vertex degrees, is studied for certain dendrimers. In this study, we present exact expressions for the F-index and F-polynomial of six infinite classes of nanostar dendrimers

Reformulated First Zagreb Index of Some Graph Operations

The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-structures

Modified Eccentric Connectivity of Generalized Thorn Graphs

The thorn graph GT of a given graph G is obtained by attaching t(>0) pendent vertices to each vertex of G. The pendent edges, called thorns of GT, can be treated as P2 or K2, so that a thorn graph is generalized by replacing P2 by Pm and K2 by Kp and the respective generalizations are denoted by GPm and GKp. The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph GT and the generalized thorn graphs GKp and GPm