## Abstract

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A_{α}(G) matrix is the convex combinations of D(G) and A(G), and is defined as A_{α}(G) = αD(G)+(1 − α)A(G), for 0 ≤ α ≤ 1. Let ρ1 ≥ ρ2 ≥ … ≥ ρn be the eigenvalues of Aα(G) (which we call α -adjacency eigenvalues of the graph G). The generalized adjacency energy also called α -adjacency energy of the graph G is defined as E_{Aα} (G) = (Formula presented.), where (Formula presented.) n is the average vertex degree, m is the size and n is the order of G. The α -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as E^{A0} (G) = E (G) and 2E ^{A1/2}(G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds.

Original language | English |
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Pages (from-to) | 127-141 |

Number of pages | 15 |

Journal | Journal of Mathematical Inequalities |

Volume | 18 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2024 |

## Keywords

- Adjacency matrix
- Laplacian (signless Laplacian) matrix
- regular graph
- α - adjacency matrix
- α -adjacency energy

## ASJC Scopus subject areas

- Analysis