We present observations in CO ( 3-2 ) that , combined with previous observations in CO ( 2-1 ) , constrain the physical properties of the filamentary molecular gas in the central \sim 6.5 kpc of ( catalog NGC 1275 ) , the central giant elliptical galaxy of the Perseus cluster . We find this molecular gas to have a temperature \gtrsim 20 K and a density \sim 10 ^ { 2 } – 10 ^ { 4 } { cm ^ { -3 } } , typically warmer and denser than the bulk of Giant Molecular Clouds ( GMCs ) in the Galaxy . Bathed in the harsh radiation and particle field of the surrounding intracluster X-ray gas , the molecular gas likely has a much higher ionization fraction than that of GMCs . For an ionization fraction of \sim 10 ^ { -4 } , similar to that of Galactic diffuse ( \lesssim 250 { cm ^ { -3 } } ) partially-molecular clouds that emit in HCN ( 1-0 ) and HCO ^ { + } ( 1-0 ) , we show that the same gas traced in CO can produce the previously reported emissions in HCN ( 3-2 ) , HCO ^ { + } ( 3-2 ) , and CN ( 2-1 ) from ( catalog NGC 1275 ) ; the dominant source of excitation for all the latter molecules is collisions with electrons . To prevent collapse , as evidenced by the lack of star formation in the molecular filaments , they must consist of thin strands that have cross-sectional radii \lesssim 0.2–2 pc if supported solely by thermal gas pressure ; larger radii are permissible if turbulence or poloidal magnetic fields provide additional pressure support . We point out that the conditions required to relate CO luminosities to molecular gas masses in our Galaxy are unlikely to apply in cluster central elliptical galaxies . Rather than being virialized structures analogous to GMCs , we propose that the molecular gas in ( catalog NGC 1275 ) comprises pressure-confined structures created by turbulent flows .