The tire structure is a key element for the tire behavior.

To make a fair comparison between a wider and narrower tire, the tire structure should be the same.

The second element is pressure. On perfect surface; higher pressure reduces the rolling resistance. But on bad surface like off-road its the opposite. A real average road is somewhere in between. To make a fair comparison, the pressure should be the same.

With the same structure and pressure, it has been demonstrated that a wider tire has lower rolling resistance.

The overall diameter has also an influence, a higher diameter reduces the rolling resistance but, as already explained, between ETRTO406 with a wider tire and ETRTO451 with a narrower tire, the difference in overall diameter is very small, too small to have a real influence.

The inner tire has also an influence on rolling resistance, not only for its weight, but also because it influences the deformation of the tire. Latex and TPU inner tubes give a lower rolling resistance than Butyl inner tubes.

Besides rolling resistance, there is also air resistance: long time ago, it was considered that narrower tire were better, less resistance. But nowadays, things changed, and we know that its a combination of the tire and rim: with matching tire and rim, its possible to have as low air resistance with a wider tire. The shape of the wheel side where tire and rim joint is a key element for the wheel aerodynamics, a good shape will reduce turbulence's. Its the reason why bicycle rim width is also increasing.

Of course, to make any valid comparison between tires and wheels, the same bike should be used.

Schwalbe has a short but clear explanation about tire rolling resistance: https://www.schwalbe.com/en/technolo...ng-resistance/

I also made some own comparison af several wheel sizes, tire width and tire types on my Birdy Titanium that allow several wheel size because it has disc brakes. I tested 3 wheel sizes and 3 types of tires:
- ETRTO349 with Continental Contact Urban 35x349 and Greenspeed Scorcher 40x349.
- ETRTO355 with Big Apple 50x355 (because there are no Big Apple nor any other 50mm wide tire in ETRTO349).
- ETRTO406 with Continental Contact Urban 32x406 (there are also wider tires in ETRTO406 including Scorcher 40x406 but the Birdy frame does not accept tires wider than about 35mm in ETRTO406).

The faster configuration was the 40x349 with Greenspeed Scorcher. The 32x406 maybe rolls better than the 35x349 Conti Contact Urban but the bigger wheels had more inertia and this was more noticeable than the lower rolling resistance.

Between the 35x349, 50x355 and 32x406, the faster and most pleasant to ride on average to bad surface was the 50x355 due to the wider tire and lower pressure. The two Conti's were much less comfortable than the Big Apple.
I am now using mainly the Big Apple wheelset because the Greenspeed Scorcher are slick pure road tires and are not usable on trails, too fragile for trails with stones, slippery on wet trails.
The ETRTO406 wheels were the less pleasant to ride due to the higher inertia that changes the Birdy behavior (I must also say that I do not fully understand why the Birdy behavior is so different between the 32x406 and 50x355 since both have almost the same overall diameter and the Big Apple are heavier than the Conti's ???).

The 3 wheelsets for the tests were very similar, same hubs (Hubsmith front, Hope rear with SRAM XDR freewheel), same type of spokes (Sapim CX-ray), similar rims (I could not find the same type of rims in the three rim sizes, of course, the ETRTO406 rim was slightly heavier and spokes longer so also slightly heavier), same cassette (Ethirtheen).