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The Number of Spanning Trees for The Generalized Cones of $K_n$, The Generalized Half Cones of $K_{m,n}$ and Some Family of Modified $K_{m,n}$

Zubeyir Cinkir

Abstract

We compute the total number of spanning trees for the generalized cone of the complete graph $K_n$ and a number of families of some modified bipartite graphs $K_{m,n}$. In particular, we obtain a new method of finding the number of spanning trees of $K_n$ and $K_{m,n}$. Our method relies on the vertex deletion formula for the number of spanning trees.

The Number of Spanning Trees for The Generalized Cones of $K_n$, The Generalized Half Cones of $K_{m,n}$ and Some Family of Modified $K_{m,n}$

Abstract

We compute the total number of spanning trees for the generalized cone of the complete graph and a number of families of some modified bipartite graphs . In particular, we obtain a new method of finding the number of spanning trees of and . Our method relies on the vertex deletion formula for the number of spanning trees.

Paper Structure

This paper contains 5 sections, 9 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalized Cones of Complete Graphs
  3. Modified Complete Bipartite Graphs
  4. Generalized Complete Bipartite Graphs
  5. Generalized Half Cone of $M^{k_1,k_2,\ldots, k_m} K_{m,n}$

Key Result

Theorem 1.1

C1 Let $u \in V(G)$, $N_G(u)= \{ p_1, \, \ldots, \, p_n \} \subset V(G)$ for a graph $G$, and let $u$ be adjacent to the vertex $p_i$ via by $a_i \geq 1$ number of edges for each $i \in \{1, \, \ldots, \, n \}$ with $n \geq 2$. If $u$ is not a cut vertex, then for $G$ and $H=G-u$ we have where $I_S$ is the set of indexes of the vertices in $S$, and $H_S$ is the graph obtained from $H$ by identify

Figures (2)

  • Figure 1: The graphs $C^3K_3$ and $M^2K_{3,4}$.
  • Figure 2: The graphs $M^{3,2}K_{2,3}$, $F^2M^{1,1}K_{2,3}$ and $F^2M^{1,3}K_{2,3}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 5 more