1. Carnegie Mellon → Berkeley →
X-Ramanujan Graphs
Sidhanth Mohanty
Carnegie Mellon → Berkeley →
Ryan O’Donnell
Carnegie Mellon

2. Let X be an infinite graph. (Of bounded max-degree.)

3. Let X be an infinite graph. (Of bounded max-degree.)
X = ℤ2⁎ℤ2⁎ℤ2 = 𝕋3, the infinite 3-regular tree

4. Let X be an infinite graph. (Of bounded max-degree.)
X = ℤ2⁎ℤ2⁎ℤ2⁎ℤ2 = 𝕋4, the infinite 4-regular tree

5. Let X be an infinite graph. (Of bounded max-degree.)
X = 𝕋3,4, the infinite (3,4)-biregular tree

6. Let X be an infinite graph. (Of bounded max-degree.)
X = ℤ3⁎ℤ2, Cayley graph of the modular group PSL2(ℤ)

7. Let X be an infinite graph. (Of bounded max-degree.)

8. Let X be an infinite graph. (Of bounded max-degree.)

9. Let X be an infinite graph. (Of bounded max-degree.)

10. Let X be an infinite graph. (Of bounded max-degree.)

11. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Here: “X covers G” (or “G is a quotient of X”) if there is a
surjection γ : X → G that preserves incidences and is a “local bijection” (on edge-neighborhoods).
Informally: G “locally resembles” X.

12. { 4-regular graphs where every vertex participates in a 3-cycle }
FinQuo(X) =
{ 4-regular graphs where every vertex participates in a 3-cycle }
X = ℤ3⁎ℤ2⁎ℤ2

13. G
FinQuo(𝕋4) =
{ 4-regular graphs }
X = 𝕋4

14. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Fact: If X is Δ-regular, this common λ1(G) is Δ.
Fact: (The “trivial eigenvalue”.)
λ1(G) ≥ λ2(G) ≥ λ3(G) ≥ ··· ≥ λn(G) are the eigenvalues of G’s adjacency matrix.

15. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Fact: If X is Δ-regular, this common λ1(G) is Δ.
Fact: (The “trivial eigenvalue”.)
We are interested in how small λ2(G) can be, for G ∈ FinQuo(X).
Why? Gap λ1(G) − λ2(G) measures how good an “expander” G is.
(You might also worry about λ1(G) − |λn(G)|, but we won’t.)

16. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Want: A statement like, “There are infinitely many G ∈ FinQuo(X) with λ2(G) ≤ ρ*.”
What’s the best ρ* that could be there?
Alon–Boppana ’90, Grigorchuk–Żuk ’99 Theorem: You couldn’t put anything smaller than ρ(X) = SpectralRadius(X).

17. k Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Want: A statement like, “There are infinitely many G ∈ FinQuo(X) with λ2(G) ≤ ρ*.”
Classic Alon–Boppana case: FinQuo(𝕋Δ) = { Δ-regular graphs }
ρ(𝕋Δ) = 2 Δ−1
“Any n-vertex Δ-regular graph has λ2(G) ≥ 2 Δ−1 − on(1) .”
Serre: Indeed, has λ (G) ≥ 2 Δ−1 − on(1) for any fixed k. k

18. k Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Want: A statement like, “There are infinitely many G ∈ FinQuo(X) with λ2(G) ≤ ρ*.”
Classic Alon–Boppana case: FinQuo(𝕋Δ) = { Δ-regular graphs }
ρ(𝕋Δ) = 2 Δ−1
“Any n-vertex Δ-regular graph has λ2(G) ≥ 2 Δ−1 − on(1) .”
Serre: Indeed, has λ (G) ≥ 2 Δ−1 − on(1) for k = Ω(n). i k

19. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Want: A statement like, “There are infinitely many G ∈ FinQuo(X) with λ2(G) ≤ ρ*.”
Grigorchuk–Żuk general case:
“Any n-vertex G ∈ FinQuo(X) has λ2(G) ≥ ρ(X) − on(1) .”
Indeed, has λk(G) ≥ ρ(X) − on(1) for k = Ω(n).

20. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Want: A statement like, “There are infinitely many G ∈ FinQuo(X) with λ2(G) ≤ ρ*.”
[AB]: “Any n-vertex Δ-regular graph has λ2(G) ≥ 2 Δ−1 − on(1) .”
Traditional Def: Δ-regular G is Ramanujan if λ2(G) ≤ 2 Δ−1 .
(An “optimal spectral expander”.)

21. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Theorem: All n-vertex G ∈ FinQuo(X) have λ2(G) ≥ ρ(X) − on(1).
Definition: G ∈ FinQuo(X) is X-Ramanujan if λ2(G) ≤ ρ(X).
[Lubotzky–Phillips–Sarnak, Margulis ’88]: There are infinitely many 𝕋Δ-Ramanujan graphs
if Δ−1 is a prime
power [Morgenstern ’94]
[Marcus–Spielman–Srivastava ’15] via the “Interlacing Polynomials Method”

22. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Theorem: All n-vertex G ∈ FinQuo(X) have λ2(G) ≥ ρ(X) − on(1).
Definition: G ∈ FinQuo(X) is X-Ramanujan if λ2(G) ≤ ρ(X).
[MSS ’15]: For X a (universal cover) tree and G0 ∈ FinQuo(X), infinitely many “lifts” G of G0 (all in FinQuo(X)) with max { Eigs(G) \ Eigs(G0) } ≤ ρ(X).
(Proved conjectures of [Bilu–Linial, Clark ’06].)

23. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Theorem: All n-vertex G ∈ FinQuo(X) have λ2(G) ≥ ρ(X) − on(1).
Definition: G ∈ FinQuo(X) is X-Ramanujan if λ2(G) ≤ ρ(X).
[MSS ’15]: For X a (universal cover) tree and G0 ∈ FinQuo(X), infinitely many “lifts” G of G0 (all in FinQuo(X)) with max { Eigs(G) \ Eigs(G0) } ≤ ρ(X).
⇒ if ∃ a single X-Ramanujan G0, there are infinitely many
⇒ ∃ infinitely many “k0-quasi-X-Ramanujan” graphs G

24. Let X be an infinite graph. (Of bounded max-degree.)
Definition: Let FinQuo(X) be all finite graphs “covered” by X.
Fact: All G ∈ FinQuo(X) have the same λ1(G).
Theorem: All n-vertex G ∈ FinQuo(X) have λ2(G) ≥ ρ(X) − on(1).
Definition: G ∈ FinQuo(X) is X-Ramanujan if λ2(G) ≤ ρ(X).
[MSS ’15]: For X a (universal cover) tree and G0 ∈ FinQuo(X), infinitely many “lifts” G of G0 (all in FinQuo(X)) with max { Eigs(G) \ Eigs(G0) } ≤ ρ(X).
Our Thm: Same holds for any “additive product graph” X.

25. Usual random “lifts” produce graphs that locally resemble trees.
So we invent a new kind: “additive lifts”.

26. Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set. A1 w v u w v u A3 w v u A2 w v u A4

27. + + + Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set. A1 w v u w v u A2 w v u A3 w v u A4 + + +
Operation #1: Summing

28. + + + Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set. A1 w v u w v u A2 w v u A3 w v u A4 + + + w v u
Operation #1: Summing
A1 + A2 + A3 + ∙∙∙ + Ac =

29. Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set.
Operation #2: Balanced r-fold lift of a single atom A1 w v u

30. Random “Additive Lifts”
Operation #2: Balanced r-fold lift of a single atom w v u

31. Random “Additive Lifts”
Operation #2: Balanced r-fold lift of a single atom
Make r copies of each vertex w v u

32. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2 u3 v3 w3 w v u u4 v4 w4

33. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
(132) v3 w3 w v u u4 id
(143) v4 w4

34. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to
(132) v3 w3 w v u u4 id
(143) v4 w4

35. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3 u
(132)
(14)(23) u4 v w
(134) v4 w4

36. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3 u
(132)
(14)(23) u4 v w
(134) v4 w4

37. Random “Additive Lifts”u1 v1 w1
Operation #2: Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3 u
(132)
(14)(23) u4
Remark: In “usual” lifts, edges get independent (random) permutations. v w
(134) v4 w4

38. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3 w v u
Results in r disjoint copies of base graph u4 v4 w4

39. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

40. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

41. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

42. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

43. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

44. Random “Additive Lifts”u1 v1 w1
Balanced r-fold lift of a single atom u2
Make r copies of each vertex v2 w2
Put a (random) perm. πu ∈ 𝔖r on each vertex u3
For each original edge (u,v), put matching between u×[r], v×[r], corresp. to v3 w3
Results in r disjoint copies of base graph u4 v4 w4

45. Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set. A1 w v u w v u A2 w v u A3 w v u A4
Random Additive r-Fold Lift of (A1, A2, A3, …, Ac):
Do a random balanced r-fold lift of each atom A1, A2, A3, …, Ac
and sum the results.

46. v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4

47. v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4 v4 v3 v2 v1 u4 u3 u2 u1 w2 w1 w3 w4

49. Random “Additive Lifts”
Let A1, A2, A3, …, Ac be “atom” graphs on the same vertex set. A1 w v u w v u A2 w v u A3 w v u A4
Random Additive r-Fold Lift of (A1, A2, A3, …, Ac):
Do a random balanced r-fold lift of each atom A1, A2, A3, …, Ac
and sum the results.

50. = + + + Random “Additive Lifts” generalize random “usual” lifts w v ux w v u x u x u x u x = + + + v w v w v w
Write the base graph as the sum of its “edge-atoms”.
Random Additive r-Fold Lift of (A1, A2, A3, …, Ac):
Do a random balanced r-fold lift of each atom A1, A2, A3, …, Ac
and sum the results.

51. 4-fold additive lift 10-fold additive lift A1 w v u w v u A2 w v u A3

52. The “additive product” jw v u w v u A2 w v u A3 w v u A4
Question:
As r → ∞, what infinite graph do random r-fold additive lifts “locally resemble”?
Answer:
The “additive product” j A1 A2 A3 A4

53. A1 w v u w v u A2 w v u A3 w v u A4 v w u A1 A2 A3 A4

54. w v u x u x w v u x = v w

55. w v u x w v u x w v u x w v u x w v u x w v u x w v u x =

56. w v u x = w v u x w v u x w v u x w v u x w v u x w v u x

57. w v u x
Fact: Doing this for any base G0 gives the universal cover tree of G0. w v u x w v u x w v u x w v u x w v u x w v u x =

58. = Fact: Every* additive lift G of (A1, A2, …, Ac) is in FinQuo( ). A1
∙∙∙ Ac
*Provided it’s connected. w v u x w v u x w v u x =

59. Fact: Every* additive lift G of (A1, A2, …, Ac)
is in FinQuo( ). A1 A2
∙∙∙ Ac

60. Fact: Every* additive lift G of (A1, A2, …, Ac)
is in FinQuo( ). A1 A2
∙∙∙ Ac
Fact: Eigs(G) = Eigs(A1+A2+∙∙∙+Ac)
∪ “new eigenvalues”
Dream: For a random additive lift G,
w.h.p. max{new eigs} ≤ ρ( ) A1 A2
∙∙∙ Ac
Would imply our theorem, that for any X = , there are infinitely many (quasi-)X-Ramanujan graphs. A1 A2
∙∙∙ Ac

61. Fact: Every* additive lift G of (A1, A2, …, Ac)
is in FinQuo( ). A1 A2
∙∙∙ Ac
Fact: Eigs(G) = Eigs(A1+A2+∙∙∙+Ac)
∪ “new eigenvalues”
Theorem: For a random additive lift G,
w.h.p. max{new eigs} ≤ ρ( )
holds with positive probability.
(As in [MSS]’s Interlacing Polynomials Method.) A1 A2
∙∙∙ Ac

62. Fact: Every* additive lift G of (A1, A2, …, Ac)
is in FinQuo( ). A1 A2
∙∙∙ Ac
Fact: Eigs(G) = Eigs(A1+A2+∙∙∙+Ac)
∪ “new eigenvalues”
Theorem: For a random additive lift G,
w.h.p. max{new eigs} ≤ ρ( )
holds with positive probability. A1 A2
∙∙∙ Ac
As in [Hall–Puder–Sawin ’18], we prove this for random r-fold additive lifts for any r.

63. Fact: Every* additive lift G of (A1, A2, …, Ac)
is in FinQuo( ). A1 A2
∙∙∙ Ac
Fact: Eigs(G) = Eigs(A1+A2+∙∙∙+Ac)
∪ “new eigenvalues”
Theorem: For a random additive lift G,
w.h.p. max{new eigs} ≤ ρ( )
holds with positive probability. A1 A2
∙∙∙ Ac
But due to lack of time in this talk, I’ll focus just on 2-fold lifts. (Then can do repeated 2-fold lifts, as in [MSS#1].)

64. Fact: New eigs of a random additive 2-lift of (A1, A2, …, Ac)
≡ Eigs of sum of random balanced edge-signings of atoms. +1 −1 −1 −1 +1

65. Fact: New eigs of a random additive 2-lift of (A1, A2, …, Ac)
≡ Eigs of sum of random balanced edge-signings of atoms. −1 +1 +1 −1 +1

66. + + Fact: New eigs of a random additive 2-lift of (A1, A2, …, Ac)
≡ Eigs of sum of random balanced edge-signings of atoms. +1 +1 −1 + +1 + +1 −1 +1 +1 +1 −1 +1 −1 +1 +1 +1 −1 +1 −1 +1

67. + + Fact: New eigs of a random additive 2-lift of (A1, A2, …, Ac)
≡ Eigs of sum of random balanced edge-signings of atoms. +1 +1 −1 + +1 + +1 −1 +1 +1 +1 −1 +1 −1 +1 +1 +3 −1

68. Fact: New eigs of a random additive 2-lift of (A1, A2, …, Ac)
≡ Eigs of sum of random balanced edge-signings of atoms. +1 −1 −1 −1 +1
The new weighted adjacency matrix is
Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc
where Qi’s are random diagonal ±1 matrices. +3 −1

69. The new weighted adjacency matrix is Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc
where Qi’s are random diagonal ±1 matrices.
max { Eigs(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) } ≤ ρ( ).
Want to show: With positive probability, A1 A2
∙∙∙ Ac
max { Eigs(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) }

70. The new weighted adjacency matrix is Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc
where Qi’s are random diagonal ±1 matrices.
max { Eigs(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) } ≤ ρ( ).
Want to show: With positive probability, A1 A2
∙∙∙ Ac
maxroot {φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) }
(where φ(∙) denotes the characteristic polynomial)
By the Interlacing Polynomials Method [MSS#4, HPS], it ultimately suffices to show A1 A2
∙∙∙ Ac
maxroot E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
≤ ρ( ).

71. E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
By the Interlacing Polynomials Method [MSS#4, HPS], it ultimately suffices to show A1 A2
∙∙∙ Ac
maxroot E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcA2Qc) ]
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
≤ ρ( ).

72. E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
Definition: is the
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
additive characteristic polynomial α(A1, A2, ∙∙∙, Ac) of (A1, A2, ∙∙∙, Ac).
Remark: The Interlacing Polynomials stuff establishes that α(A1, A2, ∙∙∙, Ac) is always a real-rooted polynomial.
In the rest of the talk we won’t need that, though. We’ll show…
max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } ≤ ρ( ) A1 A2
∙∙∙ Ac
By the Interlacing Polynomials Method [MSS#4, HPS], it ultimately suffices to show A1 A2
∙∙∙ Ac
maxroot E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcA2Qc) ]
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
≤ ρ( ).

73. E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
Definition: is the
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
additive characteristic polynomial α(A1, A2, ∙∙∙, Ac) of (A1, A2, ∙∙∙, Ac).
max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } ≤ ρ( ) A1 A2
∙∙∙ Ac

74. E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
Definition: is the
E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
additive characteristic polynomial α(A1, A2, ∙∙∙, Ac) of (A1, A2, ∙∙∙, Ac).
Special case 1: Only c = 1 atom.
α(A1) =
φ(A1), the characteristic poly
Special case 2: A1, …, Ac are the “edge-atoms” of graph G.
α(A1, …, Ac) =
μ(G), the “matching poly”
[Godsil–Gutman ’81]: μ(G) = E[φ(random edge-signing of G)]

75. [Godsil–Gutman ’81]: μ(G) = E[φ(random edge-signing of G)]
Usual def: μ(G; x) = , where
Proof of [GG]:
By expanding the determinant, , where
For A an edge-signed adj. matrix of G, only cycles in G count.
Expectation over random signing → only the 2-cycles survive.

76. [Us]: α(A1, A2, ∙∙∙, Ac) = E[ φ(Q1A1Q1+Q2A2Q2+∙∙∙+QcAcQc) ]
Similar proof ⇒ α(A1, A2, ∙∙∙, Ac; x) = , where
Compare: μ(G; x) = ,
where

77. Characteristic polynomial φ(G)
Matching polynomial μ(G)
Additive characteristic polynomial α(A1, A2, …, Ac)
“Trace Method”: kth power sum of roots(φ(G))
equals number of closed length-k walks in G.
Provable using generatingfunctionology from
Because “closed walk” = “cycles attached in a treelike fashion.”

78. Characteristic polynomial φ(G)
Matching polynomial μ(G)
Additive characteristic polynomial α(A1, A2, …, Ac)
[Godsil ’81]: kth power sum of roots(μ(G))
equals number of closed length-k “treelike walks” in G.
Provable using generatingfunctionology from μ
Because “treelike walk” = “edges attached in a treelike fashion.”

79. Characteristic polynomial φ(G)
Matching polynomial μ(G)
Additive characteristic polynomial α(A1, A2, …, Ac)
[Us]: kth power sum of roots(α(A1, A2, …, Ac))
equals number of closed length-k “freelike walks” in Ai’s.
Provable using generatingfunctionology from α
“freelike walk” := “colored cycles attached in a treelike fashion.”

80. R such that # closed length-2k walks
∴ max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } = L such that
# closed length-2k freelike walks in Ai’s ≍ L2k
Goal: max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } ≤ ρ( ) A1 A2
∙∙∙ Ac
R such that # closed length-2k walks
in (from worst starting vertex) ≍ R2k A1 A2
∙∙∙ Ac
[Us]: kth power sum of roots(α(A1, A2, …, Ac))
equals number of closed length-k “freelike walks” in Ai’s.
“freelike walk” := “colored cycles attached in a treelike fashion.”

81. R such that # closed length-2k walks
∴ max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } = L such that
# closed length-2k freelike walks in Ai’s ≍ L2k
Goal: max-magnitude-root{ α(A1, A2, ∙∙∙, Ac) } ≤ ρ( ) A1 A2
∙∙∙ Ac
R such that # closed length-2k walks
in (from worst starting vertex) ≍ R2k A1 A2
∙∙∙ Ac
∴ suffices to show that distinct closed freelike walks in Ai’s
correspond to distinct closed walks in A1 A2
∙∙∙ Ac
This is obvious from the definitions. :-)

82. A1 w v u
Thanks! v w u w v u A2 w v u A3 w v u w v u A4