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2014•Unpublished venueRequires access

The Kirchhoff index of circulant graphs

Zhou Hou-qin

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Abstract

The Kirchhoff index,Kf(G),is the sum of resistance distances between all pairs of vertices in G.A graph Gis called circulant graph if it has a circulant adjacency matrix.A graph Gis called an integral graph if it has an integral spectrum.In this paper,using Laplacian spectrum of circulant graphs,we discuss and give a new lower bound for the Kirchhoff index.According to Ramanujan sums,applying the Euler function and the Mobius function,we obtain a simple formula for the Kirchhoff index of integral circulant graphs.Then we can calculate the Kirchhoff index,and we do not calculate its eigenvalues.

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What this paper is about

The Kirchhoff index,Kf(G),is the sum of resistance distances between all pairs of vertices in G.A graph Gis called circulant graph if it has a circulant adjacency matrix.A graph Gis called an integral graph if it has an integral spectrum.In this paper,using Laplacian spectrum of circulant graphs,we discuss and give a new lower bound for the Kirchhoff index.According to Ramanujan sums,applying the Euler function and the Mobius function,we obtain a simple formula for the Kirchhoff index of integral circulant graphs.Then we can calculate the Kirchhoff index,and we do not calculate its eigenvalues.

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Available abstract

The Kirchhoff index,Kf(G),is the sum of resistance distances between all pairs of vertices in G.A graph Gis called circulant graph if it has a circulant adjacency matrix.A graph Gis called an integral graph if it has an integral spectrum.In this paper,using Laplacian spectrum of circulant graphs,we discuss and give a new lower bound for the Kirchhoff index.According to Ramanujan sums,applying the Euler function and the Mobius function,we obtain a simple formula for the Kirchhoff index of integral circulant graphs.Then we can calculate the Kirchhoff index,and we do not calculate its eigenvalues.

Key concepts: Circulant matrix, Resistance distance, Ramanujan's sum, Mathematics, Laplacian matrix, Circulant graph, Adjacency matrix, Graph energy

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