Authors:
Huahui Ding, Danyi Li, Weigen Yan
Abstract
Let \( G = (V(G), E(G)) \) be a graph with adjacency matrix \( A(G) \). The energy \( \mathcal{E}(G) \) of \( G \) is defined as the sum
of the absolute values of eigenvalues of \( A(G) \).
An open problem posed by Gutman is to determine how to change the energy of graphs when an edge is deleted or added.
In this paper, we prove that, for a bipartite graph \( G \) and \( e\in E(G^c) \), if each cycle \( C_i \) of \( G+e \) containing \( e \)
has length \( |V(C_i)| \not\equiv 0\ (mod\ 4) \), then \( \mathcal{E}(G) < \mathcal{E}(G+e) \), where \( G^c \) is the complement of
\( G \). We also prove that, for a tree \( T \) and \( e=uv\in E(T^c) \), if the unique cycle \( C \) of \( T+e \) satisfies
\( |V(C)|\equiv 0\ (mod\ 4) \) and there exists a pendent vertex of \( T+e \) adjacent with one of vertices of \( C \) different
from \( u \) and \( v \), then \( \mathcal E(T)<\mathcal E(T+e) \).
