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Explicit Rayleigh's Principles for Resistive Electrical Network and The Total Number of Spanning Trees of Graphs

Zubeyir Cinkir

Abstract

We give identities for the voltage and resistance functions on a metrized graph to show how these functions behave under any edge deletion/contraction and the identification of any two vertices. This leads to explicit versions of Rayleigh's Principles on a resistive electrical network. We also establish Euler's Identities for the resistance and the voltage functions on an electrical network. One can use these results to study various invariants of metrized graphs and electrical networks. As a specific application, we obtain various identities for the total number of spanning trees of a graph. For example, we show how the total number of spanning trees changes under graph operations such as, contraction of an edge, deletion of an edge, deletion of a vertex, the join of arbitrary two or three vertices.

Explicit Rayleigh's Principles for Resistive Electrical Network and The Total Number of Spanning Trees of Graphs

Abstract

We give identities for the voltage and resistance functions on a metrized graph to show how these functions behave under any edge deletion/contraction and the identification of any two vertices. This leads to explicit versions of Rayleigh's Principles on a resistive electrical network. We also establish Euler's Identities for the resistance and the voltage functions on an electrical network. One can use these results to study various invariants of metrized graphs and electrical networks. As a specific application, we obtain various identities for the total number of spanning trees of a graph. For example, we show how the total number of spanning trees changes under graph operations such as, contraction of an edge, deletion of an edge, deletion of a vertex, the join of arbitrary two or three vertices.

Paper Structure

This paper contains 5 sections, 40 theorems, 116 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Explicit Rayleigh's Shorting Law
  3. The Deletion and the Contraction Identities for the Resistance Function (Revisited)
  4. Spanning Trees and Two Vertex Identification
  5. Spanning Trees and More Than Two Vertex Identification

Key Result

Theorem 1.1

For any $p, \, q, \, s, \, t \in V(G)$, we have In particular, ${r(s,t) \geq r_{G_{pq}}(s,t) }$ and the equality holds if and only if $j_{p}(q,s)= j_{p}(q,t)$.

Figures (9)

  • Figure 1: The graph $\Gamma$ and $\Gamma_{pq}$ after circuit reductions until four vertices.
  • Figure 2: The graph $\Gamma$ and $\Gamma_{pq}$ after circuit reductions until three vertices.
  • Figure 3: The graph $\Gamma-e_i$ after circuit reductions until four vertices and $e_i$, and obtaining ${\overline{\Gamma}}_i$.
  • Figure 4: The graph $\Gamma-e_i$ after circuit reductions until two vertices and $e_i$, and applying Delta-Y transformation.
  • Figure 5: The graphs $P_3$, $P_6$, $C_2$, $G_4$, $B_2$ and $B_3$.
  • ...and 4 more figures

Theorems & Definitions (74)

  • Theorem 1.1: Explicit Rayleigh’s Shorting Law for graphs
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Explicit Rayleigh’s Shorting Law
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5: Magical Identities
  • Remark 2.6
  • Theorem 2.7
  • ...and 64 more