Searches of exotic Higgs bosons in general mass spectra of the Georgi-Machacek model at the LHCCheng-Wei Chiang1,2,3, An-Li Kuo1*, Toshifumi Yamada1,41Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli, Taiwan2Institute of Physics, Academia Sinica, Taipei, Taiwan3Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan4School of Physics, Korea Institute for Advanced Study, Seoul, Korea* presenting author:An-Li Kuo, email:gescheitnie@gmail.comWe derive most general sets of viable mass spectra of the exotic Higgs bosons in the Georgi-Machacek model that are consistent with the theoretical constraints of vacuum stability and perturbative unitarity and the experimental constraints of electroweak precision observables, $Zb \bar b$ coupling and Higgs boson signal strengths. Branching ratios of various cascade decay channels of the doubly-charged Higgs boson in the ${\bf 5}$ representation, the singly-charged Higgs boson in ${\bf 3}$, and the singlet Higgs boson are further computed. As the most promising channel for discovering the model, we study the prospects for detecting the doubly-charged Higgs boson that is produced via the vector boson fusion process and decays into final states containing a pair of same-sign leptons at the 14-TeV LHC and a 100-TeV future $pp$ collider. For this purpose, in particular, we evaluate the acceptance times efficiency for a signal of the doubly-charged Higgs boson with general viable mass spectra, and compare it with the standard model background estimates.Keywords: Higgs Physics, Georgi-Machacek Model