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The change of vertex energy when joining trees

Octavio Arizmendi, Saylé Sigarreta

Abstract

In this manuscript we study how the vertex energy of a tree is affected when joined with a bipartite graph. We find an alternating pattern with respect to the coalescence vertex: the energy decreases for vertices located at odd distances and increases for those located at even distances.

The change of vertex energy when joining trees

Abstract

In this manuscript we study how the vertex energy of a tree is affected when joined with a bipartite graph. We find an alternating pattern with respect to the coalescence vertex: the energy decreases for vertices located at odd distances and increases for those located at even distances.
Paper Structure (9 sections, 14 theorems, 26 equations, 2 figures)

This paper contains 9 sections, 14 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions in Graph Theory
  4. Graph energy
  5. Energy of a vertex
  6. Main Results
  7. Successive coalescence
  8. Examples
  9. Conclusion

Key Result

Lemma 2.1

(Theorem 1.3, b14) Let $u v$ be an edge of $G$. Then where $\mathscr{C}(u v)$ is the set of cycles containing $u v$.

Figures (2)

  • Figure 1: Graphical representation of the behavior observed in the Theorem \ref{['t1']}.
  • Figure 2: Graphical representation of the behavior observed in the Theorem \ref{['t2']}.

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 3.2
  • proof
  • ...and 13 more