On the Normalised Laplacian Spectrum, Degree-Kirchhoff Index and Spanning Trees of Graphs

Abstract

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

Study on the normalized Laplacian of a penta‐graphene with applications

“…Chen and Zhang proposed a novel graph invariant, namely, the multiplicative degree‐Kirchhoff index Kf * ( G ), defined as Kf * ( G ) = ∑ i < j d i d j r ij (see also References ). It is motivating to observe that this novel graph invariant is closely linked with the spectrum of the ℒ( G ) (see Lemma 1.6 in the next section).…”

“…In this work, motivated by, we focus on two interesting types of molecular graphs: the penta‐graphene (penta‐C) R n and the pentagonal Möbius ring (see Figure ). The penta‐graphene (penta‐C) R n is the graph obtained from the linear pentagonal chain L n by identifying the opposite lateral edges in an ordered way, that is, penta‐graphene (penta‐C) R n is obtained from L n by identifying the vertex 1 and (2 n + 1), vertex 1 ′ and (2 n + 1) ′ , respectively.…”

On the normalized Laplacian of Möbius phenylene chain and its applications

“…Hence, it is interesting to obtain the closed‐form formula for the multiplicative degree‐Kirchhoff index of graph G . For recent advances on this topic, one may be referred to and the references with in.…”

On normalized Laplacians, multiplicative degree‐Kirchhoff indices, and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives

“…The multiplicative degree‐Kirchhoff index , written as Kf * ( G ), of a graph G was proposed in Reference . It is defined as (see References ). It is interesting to see that this novel graph invariant is closely related to the corresponding spectrum of the normalized Laplacian (see Lemma 2.2 in the next section).…”