Chromatic Zagreb indices for graphical embodiment of colour clusters

International audienceFor a colour cluster C = (C 1 , C 2 , C 3 ,. .. , C), where C i is a colour class such that |C i | = r i , a positive integer, we investigate two types of simple connected graph structures G C 1 , G C 2 which represent graphical embodiments of the colour cluster such that the chromatic numbers χ(G C 1) = χ(G C 2) = and min{ε(G C 1)} = min{ε(G C 2)} = i=1 r i − 1, and ε(G) is the size of a graph G. In this paper, we also discuss the chromatic Zagreb indices corresponding to G C 1 , G C 2

A study on kk-coalescence of two graphs

The $k$-coalescence of two graphs is obtained by merging a $k$-clique of each graph. The $A_\alpha$-matrix of a graph is the convex combination of its degree matrix and adjacency matrix. In this paper, we present some structural properties of a non-regular graph which is obtained from the $k$-coalescence of two graphs. Also, we derive the $A_\alpha$-characteristic polynomial of $k$-coalescence of two graphs and then compute the $A_\alpha$-spectra of $k$-coalescence of two complete graphs. In addition, we estimate the $A_\alpha$-energy of $k$-coalescence of two complete graphs. Furthermore, we obtain some topological indices of $k$-coalescence of two graphs, and as an application, we determine the Wiener, hyper-Wiener and Zagreb indices of Lollipop and Dumbbell graphs. From these results, we calculate the Wiener, hyper-Wiener and Zagreb indices of the organic compound 1,2-dicyclohexylethane(\ce{C_{14}H_{26}})