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(D) The wavelength of maximum absorption $(\lambda_{max})$ is inversely proportional to the energy gap between the $HOMO$ and $LUMO$ of the $\pi$ syst

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$2 < 3 < 1$

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(D) The wavelength of maximum absorption $(\lambda_{max})$ is inversely proportional to the energy gap between the $HOMO$ and $LUMO$ of the $\pi$ system,as given by $E = \frac{hc}{\lambda}$.Conjugated dienes have a smaller $HOMO$-$LUMO$ gap compared to isolated dienes,which means they absorb at a higher wavelength $(\lambda_{max})$.Structure $(2)$ is an isolated diene,so it will have the lowest $\lambda_{max}$.Structures $(1)$ and $(3)$ are conjugated dienes. Structure $(1)$ is a more substituted conjugated diene compared to $(3)$,and increased substitution in a conjugated system generally leads to a bathochromic shift (higher $\lambda_{max}$).Therefore,the order of increasing $\lambda_{max}$ is $2 < 3 < 1$.

Rank the following alkenes in order of increasing $\lambda_{max}$:
$(1)$ $A$ conjugated diene in a bicyclic system.
$(2)$ An isolated diene in a bicyclic system.
$(3)$ $A$ conjugated diene in a bicyclic system (different substitution pattern).

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Rank the following alkenes in order of increasing $\lambda_{max}$:$(1)$ $A$ conjugated diene in a bicyclic system.$(2)$ An isolated diene in a bicyclic system.$(3)$ $A$ conjugated diene in a bicyclic system (different substitution pattern).