The normalized Laplacian and related indexes of graphs with edges blew up by cliques

TL;DR

This paper studies the spectral properties and graph invariants of clique-blew up graphs, created by replacing edges with cliques, and provides formulas for key graph indexes based on the original graph.

Contribution

It introduces the clique-blew up graph construction and derives explicit spectral and index formulas in terms of the original graph G.

Findings

Characterized the normalized Laplacian spectrum of clique-blew up graphs.

Derived formulas for the multiplicative degree-Kirchhoff index, Kemeny's constant, and spanning trees.

Analyzed the spectrum and indexes of iterative clique-blew up graphs.

Abstract

In this paper, we introduce the clique-blew up graph $CL(G)$ of a given graph $G$, which is obtained from $G$ by replacing each edge of $G$ with a complete graph $K_n$. We characterize all the normalized Laplacian spectrum of the grpah $CL(G)$ in term of the given graph $G$. Based on the spectrum obtained, the formulae to calculate the multiplicative degree-Kirchhoff index, the Kemeny's constant and the number of spanning trees of $CL(G)$ are derived well. Finally, the spectrum and indexes of the clique-blew up iterative graphs are present.