Which other schools have no definitive primary rival, other than LSU?

Arkansas

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LSU has some heated rivalries but Tiger fans are often split if you ask the primary rival. And no one in the SEC would say LSU is their definitive primary.

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Trying to make LSU vs ATM the rivalry was pretty dumb. I'd still have LSU vs Arkansas as the rivalry even though it's not nearly as interesting as LSU vs Bama. Arkansas being a turd most of the time doesn't help but it's still some bragging rights.

How does (or did) Texas look at A&M in terms of a rival? I know A&M is still obsessed with the Longhorns. But does Texas just look at Oklahoma as their rival?

Maybe UGA? Their fans seem to be split between GT, Florida, and Auburn. They're kind of in the same boat as LSU in that their in-state rival left the SEC a long time ago and no longer the program they once were, although UGA and GT still play every year and at least GT is still in a major conference. I saw some UGA fans saying that there have been some rumblings about ending the annual series with them but I'm not sure how credible those rumors are.

We are their rival always have been even though we sucked for decades

Tulane has no definitive primary rival.

If you look at ND these are the schools they have played the most.

Rk Opponent From To G
? W L T Pct PS PA Diff PS PA Diff
1 Navy 1927 2023 96 82 13 1 .859 2782 1204 1578 29.0 12.5 16.4
2 Southern California 1926 2023 94 51 38 5 .569 1965 1813 152 20.9 19.3 1.6
3 Purdue 1899 2021 84 57 25 2 .690 2078 1383 695 24.7 16.5 8.3
4 Pittsburgh 1930 2023 70 49 21 0 .700 2038 1063 975 29.1 15.2 13.9
5 Michigan State 1918 2017 65 36 28 1 .562 1330 1227 103 20.5 18.9 1.6
6 Army 1913 2016 49 37 8 4 .796 885 424 461 18.1 8.7 9.4
7 Northwestern 1899 2018 47 37 9 1 .798 1072 411 661 22.8 8.7 14.1
8 Michigan 1899 2019 38 17 20 1 .461 753 864 -111 19.8 22.7 -2.9
9 Georgia Tech 1922 2021 37 30 6 1 .824 926 389 537 25.0 10.5 14.5
10 Stanford 1924 2023 37 23 14 0 .622 1072 778 294 29.0 21.0

penn state

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Which other schools have no definitive primary rival, other than LSU?

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If you're talking about schools where their biggest rival doesn't also consider them their biggest rival

Arkansas (don't really have a primary rival)
Tennessee (Alabama's primary rival is Auburn)
Georgia (Auburn/FL have instate rivals that are bigger deals)
Texas A&M (Texas' primary rival is Oklahoma)
Vandy (Tennessee's primary rival is Alabama)
Missouri (they don't even play their rival anymore)

That's just the SEC

Texas Tech