Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectrum of generalized H-join operation constrained by indexing maps -- I

R. Ganeshbabu, G. Arunkumar

Abstract

Fix $m \in \mathbb N$. A new generalization of the $H$-join operation of a family of graphs $\{G_1, G_2, \dots, G_k\}$ constrained by indexing maps $I_1,I_2,\dots,I_k$ is introduced as $H_m$-join of graphs, where the maps $I_i:V(G_i)$ to $[m]$. Various spectra, including adjacency, Laplacian, and signless Laplacian spectra, of any graph $G$, which is a $H_m$-join of graphs is obtained by introducing the concept of $E$-main eigenvalues. More precisely, we deduce that in the case of adjacency spectra, there is an associated matrix $E_i$ of the graph $G_i$ such that a $E_i$-non-main eigenvalue of multiplicity $m_i$ of $A(G_i)$ carry forward as an eigenvalue for $A(G)$ with the same multiplicity $m_i$, while an $E_i$-main eigenvalue of multiplicity $m_i$ carry forward as an eigenvalue of $G$ with multiplicity at least $m_i - m$. As a corollary, the universal adjacency spectra of some families of graphs is obtained by realizing them as $H_m$-joins of graphs. As an application, infinite families of cospectral families of graphs are found.

On the spectrum of generalized H-join operation constrained by indexing maps -- I

Abstract

Fix . A new generalization of the -join operation of a family of graphs constrained by indexing maps is introduced as -join of graphs, where the maps to . Various spectra, including adjacency, Laplacian, and signless Laplacian spectra, of any graph , which is a -join of graphs is obtained by introducing the concept of -main eigenvalues. More precisely, we deduce that in the case of adjacency spectra, there is an associated matrix of the graph such that a -non-main eigenvalue of multiplicity of carry forward as an eigenvalue for with the same multiplicity , while an -main eigenvalue of multiplicity carry forward as an eigenvalue of with multiplicity at least . As a corollary, the universal adjacency spectra of some families of graphs is obtained by realizing them as -joins of graphs. As an application, infinite families of cospectral families of graphs are found.
Paper Structure (15 sections, 14 theorems, 62 equations, 4 figures)

This paper contains 15 sections, 14 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some basic definitions and lemmas
  3. $H_m$-join of graphs
  4. Various graph families as $H_m$-joins.
  5. Adjacency matrix of $H$-join and $H_m$-join of graphs
  6. $E$-Main eigenvalues of a graph
  7. Main Result
  8. Proof of the main theorem
  9. Reducing the value of $m$
  10. Universal Spectra of $H_m$-joins
  11. Applications to $H$-generalized Join of Graphs
  12. $H$-generalized join as $H_m$-join
  13. Universal spectrum of $H$-generalized join of graphs
  14. Construction of co-spectral graphs
  15. Future Directions:

Key Result

Theorem 1

Let $M_i$ be a complex matrix of order $n_i$ and let $u_i$ and $v_i$ be arbitrary complex vectors of size $n_i \times 1$ for $1 \le i \le k$. Let $n= \Sigma_{i=1}^k n_i.$ Let $\rho_{i,j}$ be arbitrary complex numbers for $1 \le i,j \le k$ and $i \ne j$. For each $1 \le i \le k$, let $\phi_i(\lambda) Then the characteristic polynomial of $A(\bold M, \bold u, \rho)$ is given by

Figures (4)

  • Figure 1: $(P_4)_5$-join of $K_3,P_4,C_5,K_{3,3}$
  • Figure 2: $(P_2)_2$-join of $\{K_2, K_5\}$
  • Figure 3: $(P_3)_3$-join of $\{K_2, P_3, K_{1,3}\}$
  • Figure 4: $(P_4)_5$- generalized join of $K_3,P_4,C_5,K_{3,3}$

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 1
  • Remark 1
  • Lemma 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Lemma 2
  • ...and 20 more