The conjugate characteristic polynomial of some compound graphs

Abstract

Let G be a simple graph of order n and let S be any subset of the vertex set V(G). We denote by G _{S, T} the graph obtained from G by adding two new non-adjacent vertices x, y, so that x is adjacent exactly to the vertices from S, and y is adjacent exactly to the vertices from T, where T=V(G)∖S. We here determine the conjugate characteristic polynomial of the graph G _{S, T} . We also consider some other compound graphs obtained from G and determine their conjugate characteristic polynomials.

Keywords:

• graph
• conjugate adjacency matrix
• conjugate characteristic polynomial

1991 AMS Subject Classification :

• 05 C 50

Acknowledgements

The author is very grateful to the reviewers and editors for their valuable comments and suggestions.

Notes

In the proof of Proposition 5, we use the fact that .

In the proof of Proposition 7, we use the fact that .