If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ v$ is a vertex in $G$. But it is not known that how the energy of a graph and its subgraphs are related if the number of vertices are same in both the graphs.
I am trying to formulate increasing or decreasing pattern of energy of a graph and its subsequent subgraphs on the same set of vertices. For instance, I am taking a complete graph $G=K_4$ with four vertices. Then I have the following observations.
(1) $E(G)=6$
(2) $E(G_1)=5.1231056256=1+\sqrt{17}$, where $G_1$ is a subgraph obtained by deleting one edge from $G$.
Similarly,
(3) $E(G_2)=4=2\sqrt{4}$ (obtained by deleting two edges from $G$)
(4) $E(G_3)=4.472135955=2\sqrt{5}$ (obtained by deleting three edges from $G$)
(5) $E(G_4)=2\sqrt{2}$ (obtained by deleting four edges from $G$)
(6) $G_5=2=2\sqrt{1}$ (obtained by deleting five edges from $G$).
I would really appreciate if someone could help me arriving at a generalizing formula of the above pattern in particular. That is, what is the general pattern in which the energy is increasing or decreasing in the above example?
