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Construction of non-regular $A_α$-cospectral graphs from some join of graphs

Najiya V K, Chithra A

Abstract

Cospectral graphs are a fascinating concept in graph theory, where two non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we compute the $A_α$-characteristic polynomial of neighbour and non-neighbour splitting join, neighbour and non-neighbour shadow join, central vertex and edge join and duplicate join of two graphs. In addition, when $\graphene_1$ and $\graphene_2$ are regular, we compute the $A_α$-spectrum of these graphs. As an application, we construct non-regular, non-isomorphic graphs that are $A_α$-cospectral.

Construction of non-regular $A_α$-cospectral graphs from some join of graphs

Abstract

Cospectral graphs are a fascinating concept in graph theory, where two non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we compute the -characteristic polynomial of neighbour and non-neighbour splitting join, neighbour and non-neighbour shadow join, central vertex and edge join and duplicate join of two graphs. In addition, when and are regular, we compute the -spectrum of these graphs. As an application, we construct non-regular, non-isomorphic graphs that are -cospectral.
Paper Structure (8 sections, 26 theorems, 28 equations, 6 figures, 1 table)

This paper contains 8 sections, 26 theorems, 28 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Splitting joins
  5. Shadow joins
  6. Central joins
  7. Duplicate join
  8. Conclusion

Key Result

Lemma 2.1

zhang2006schur If $D$ is invertible then, and if $A$ is invertible then,

Figures (6)

  • Figure 1: Non-isomorphic $A$-cospectral regular graphs
  • Figure 2: $P_4\doublesqcup P_2$
  • Figure 3: $P_4\doublesqcap P_2$
  • Figure 4: $P_4\bowtie P_2$
  • Figure 5: Non-isomorphic non-regular $A_\alpha$-cospectral graphs
  • ...and 1 more figures

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 33 more