<p>Let <i>G</i> be a connected graph. The resistance distance between any two vertices of <i>G</i> is equal to the effective resistance between them in the corresponding electrical network constructed from <i>G</i> by replacing each edge with a unit resistor. The Kirchhoff index of <i>G</i> is defined as the sum of the resistance distances between all pairs of vertices. In this paper, we consider the hexagonal subdivision of graph <i>G</i>, with the resulted graphs being denoted by <i>H</i>(<i>G</i>). Using algebraic and combinatorial methods, we derive explicit formulae for the resistance distances and Kirchhoffian graph invariants (i.e., Kirchhoff index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index) of <i>H</i>(<i>G</i>). It turns out that the resistance distances of <i>H</i>(<i>G</i>) could be expressed in terms of the resistance distances of <i>G</i>, and the Kirchhoffian graph invariants of <i>H</i>(<i>G</i>) could be expressed in terms of the Kirchhoffian graph invariants and parameters of <i>G</i>. Finally, the formulaes for these graph invariants of iterated hexagonal subdivision of graphs are obtained.</p>

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Resistance distance and Kirchhoff index in hexagonal subdivision of graphs

  • Yujun Yang,
  • Can Xu,
  • Wensheng Sun

摘要

Let G be a connected graph. The resistance distance between any two vertices of G is equal to the effective resistance between them in the corresponding electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is defined as the sum of the resistance distances between all pairs of vertices. In this paper, we consider the hexagonal subdivision of graph G, with the resulted graphs being denoted by H(G). Using algebraic and combinatorial methods, we derive explicit formulae for the resistance distances and Kirchhoffian graph invariants (i.e., Kirchhoff index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index) of H(G). It turns out that the resistance distances of H(G) could be expressed in terms of the resistance distances of G, and the Kirchhoffian graph invariants of H(G) could be expressed in terms of the Kirchhoffian graph invariants and parameters of G. Finally, the formulaes for these graph invariants of iterated hexagonal subdivision of graphs are obtained.