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ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

Part of: Circuits, networks Graph theory

Published online by Cambridge University Press:  26 February 2015

JING HUANG  and
SHUCHAO LI
JING HUANG
Affiliation:
Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China email [email protected]
SHUCHAO LI*
Affiliation:
Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China email [email protected]
*
✉[email protected]
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Abstract

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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 $q(G)$).

Keywords

resistance distanceKirchhoff indexnormalised Laplacianspanning tree

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 94C99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 353 - 367
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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