Graph spectral conditions and structural properties Hong-Jian Lai - - PowerPoint PPT Presentation

Graph spectral conditions and structural properties

Hong-Jian Lai West Virginia University

– p. 1/39

The problems

G: = a (connected) simple graph.

– p. 2/39

The problems

G: = a (connected) simple graph. AG = (aij)n×n = adjacency matrix of G. aij =    1 if i and j are adjacent if i and j are not adjacent .

– p. 2/39

The problems

G: = a (connected) simple graph. AG = (aij)n×n = adjacency matrix of G. aij =    1 if i and j are adjacent if i and j are not adjacent . The eigenvalues of AG, λ1 ≥ λ2 ≥ ... ≥ λn, are the eigenvalues of G. (spectrum of G).

– p. 2/39

The problems

G: = a (connected) simple graph. AG = (aij)n×n = adjacency matrix of G. aij =    1 if i and j are adjacent if i and j are not adjacent . The eigenvalues of AG, λ1 ≥ λ2 ≥ ... ≥ λn, are the eigenvalues of G. (spectrum of G). λ(G) = λ1(G): spectral radius of G.

– p. 2/39

The problems

Eigenvalues of G = invariants of G

– p. 3/39

The problems

Eigenvalues of G = invariants of G The Problem: Can spectral conditions of G be used to predict the structural properties of G?

– p. 3/39

The problems

• Example. Let χ(G) be the chromatic number of G.

– p. 4/39

The problems

• Example. Let χ(G) be the chromatic number of G.

Theorem (Wilf, J, London Math Soc, 1967) If G is connected, then χ(G) ≤ λ1(G) + 1,

– p. 4/39

The problems

• Example. Let χ(G) be the chromatic number of G.

Theorem (Wilf, J, London Math Soc, 1967) If G is connected, then χ(G) ≤ λ1(G) + 1, where equality holds iff G is complete or an odd cycle.

– p. 4/39

The problems

• Example. Let χ(G) be the chromatic number of G.

Theorem (Wilf, J, London Math Soc, 1967) If G is connected, then χ(G) ≤ λ1(G) + 1, where equality holds iff G is complete or an odd cycle. This has been extended to group colorings in X. K. Zhang’s dissertation (WVU 1998).

– p. 4/39

Edge-Disjoint Spanning Trees and Connectivity

κ(G): = vertex-connectivity of a graph G.

– p. 5/39

Edge-Disjoint Spanning Trees and Connectivity

κ(G): = vertex-connectivity of a graph G. κ′(G): = edge-connectivity of a graph G.

– p. 5/39

Edge-Disjoint Spanning Trees and Connectivity

κ(G): = vertex-connectivity of a graph G. κ′(G): = edge-connectivity of a graph G. τ(G): = maximum number of edge-disjoint spanning trees in G.

– p. 5/39

Edge-Disjoint Spanning Trees and Connectivity

κ(G): = vertex-connectivity of a graph G. κ′(G): = edge-connectivity of a graph G. τ(G): = maximum number of edge-disjoint spanning trees in G. Problem (Cioaba and Wong, LAA 2012): Determine the relationship between τ(G) and the eigenvalues of G.

– p. 5/39

Edge-Disjoint Spanning Trees and Connectivity

κ(G): = vertex-connectivity of a graph G. κ′(G): = edge-connectivity of a graph G. τ(G): = maximum number of edge-disjoint spanning trees in G. Problem (Cioaba and Wong, LAA 2012): Determine the relationship between τ(G) and the eigenvalues of G. Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ2(G) to warrant κ(G) ≥ k.

– p. 5/39

Edge-Disjoint Spanning Trees

Example

q q q q

• ❅

❅ ❅

– p. 6/39

Edge-Disjoint Spanning Trees

Example

q q q q

• ❅

❅ ❅

Two edge-disjoint spanning trees (τ(K4) = 2)

q q q q

• q

q q q ❅ ❅ ❅

– p. 6/39

Theorem of Nash-Williams and Tutte

[X, Y ]G: = edges of G with one end in X and the other end in Y .

– p. 7/39

Theorem of Nash-Williams and Tutte

[X, Y ]G: = edges of G with one end in X and the other end in Y . d(X) = dG(X) = |[X, V (G) − X]G|.

– p. 7/39

Theorem of Nash-Williams and Tutte

[X, Y ]G: = edges of G with one end in X and the other end in Y . d(X) = dG(X) = |[X, V (G) − X]G|. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, τ(G) ≥ k if and only if for any partition (V1, V2, ..., Vt) of V (G), 1 2

t

• i=1

d(Vi) =

• 1≤i<j≤t

|[Vi, Vj]G| ≥ k(t − 1).

– p. 7/39

Theorem of Nash-Williams and Tutte

[X, Y ]G: = edges of G with one end in X and the other end in Y . d(X) = dG(X) = |[X, V (G) − X]G|. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, τ(G) ≥ k if and only if for any partition (V1, V2, ..., Vt) of V (G), 1 2

t

• i=1

d(Vi) =

• 1≤i<j≤t

|[Vi, Vj]G| ≥ k(t − 1). There is an equivalent version of the theorem.

– p. 7/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z.

– p. 8/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z. ω(G) = number of connected component of G.

– p. 8/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z. ω(G) = number of connected component of G. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, these are equivalent.

– p. 8/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z. ω(G) = number of connected component of G. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, these are equivalent. (i) τ(G) ≥ k.

– p. 8/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z. ω(G) = number of connected component of G. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, these are equivalent. (i) τ(G) ≥ k. (ii) ∀Y ⊆ E(G), |E(G/Y )| ≥ k(|V (G/Y )| − 1).

– p. 8/39

Theorem of Nash-Williams and Tutte

If Z ⊆ E(G), then G/Z is the graph obtained from G be contracting the edges in Z. ω(G) = number of connected component of G. Theorem (Nash-Williams, Tutte [J. London Math. Soc. (1961)]) For a connected graph G, these are equivalent. (i) τ(G) ≥ k. (ii) ∀Y ⊆ E(G), |E(G/Y )| ≥ k(|V (G/Y )| − 1). (iii) ∀X ⊆ E(G), |X| ≥ k(ω(G − X) − 1).

– p. 8/39

The κ′-τ Lemma

The κ′-τ Lemma (Gusfield, IPL 1983, and Catlin, Shao, HJL DM 2009) κ′(G) ≥ 2k if and only if for any edge subset X ⊆ E(G) with |X| ≤ k, τ(G − X) ≥ k.

– p. 9/39

The κ′-τ Lemma

The κ′-τ Lemma (Gusfield, IPL 1983, and Catlin, Shao, HJL DM 2009) κ′(G) ≥ 2k if and only if for any edge subset X ⊆ E(G) with |X| ≤ k, τ(G − X) ≥ k. Sufficiency: Any edge cut must have size at least 2k.

– p. 9/39

The κ′-τ Lemma

The κ′-τ Lemma (Gusfield, IPL 1983, and Catlin, Shao, HJL DM 2009) κ′(G) ≥ 2k if and only if for any edge subset X ⊆ E(G) with |X| ≤ k, τ(G − X) ≥ k. Sufficiency: Any edge cut must have size at least 2k. Necessity: Take a partition (V1, V2, ..., Vt) of V (G − X), 2

• 1≤i<j≤t

|[Vi, Vj]G−X| =

t

• i=1

|[Vi, V − Vi]G| − 2|X| ≥ 2kt − 2k = 2k(t − 1). Then apply Nash-Williams and Tutte Theorem.

– p. 9/39

Cioaba’s Problem

Cioaba’s idea Use eigenvalues to predict edge-connectivity, then use the κ′-τ Lemma to study τ(G).

– p. 10/39

Cioaba’s Problem

Cioaba’s idea Use eigenvalues to predict edge-connectivity, then use the κ′-τ Lemma to study τ(G). Let d be an integer with 2 ≤ k ≤ d, and G be a d-regular graph.

– p. 10/39

Cioaba’s Problem

Cioaba’s idea Use eigenvalues to predict edge-connectivity, then use the κ′-τ Lemma to study τ(G). Let d be an integer with 2 ≤ k ≤ d, and G be a d-regular graph. Theorem (Cioaba, LAA 2010) If λ2(G) < d − 2(k−1)

d+1 , then

κ′(G) ≥ k.

– p. 10/39

Cioaba’s Problem

Cioaba’s idea Use eigenvalues to predict edge-connectivity, then use the κ′-τ Lemma to study τ(G). Let d be an integer with 2 ≤ k ≤ d, and G be a d-regular graph. Theorem (Cioaba, LAA 2010) If λ2(G) < d − 2(k−1)

d+1 , then

κ′(G) ≥ k. Apply The κ′-τ Lemma.

– p. 10/39

Cioaba’s Problem

Cioaba’s idea Use eigenvalues to predict edge-connectivity, then use the κ′-τ Lemma to study τ(G). Let d be an integer with 2 ≤ k ≤ d, and G be a d-regular graph. Theorem (Cioaba, LAA 2010) If λ2(G) < d − 2(k−1)

d+1 , then

κ′(G) ≥ k. Apply The κ′-τ Lemma. Corollary: (Cioaba, LAA 2010) If λ2(G) < d − 4k−2

d+1 , then

τ(G) ≥ k.

– p. 10/39

Cioaba’s Problem

Let G be a d-regular graph.

– p. 11/39

Cioaba’s Problem

Let G be a d-regular graph. Theorem (Cioaba and Wong, LAA 2012) Assume that 4 ≤ d. If λ2(G) < d −

3 d+1, then τ(G) ≥ 2.

– p. 11/39

Cioaba’s Problem

Let G be a d-regular graph. Theorem (Cioaba and Wong, LAA 2012) Assume that 4 ≤ d. If λ2(G) < d −

3 d+1, then τ(G) ≥ 2.

Theorem (Cioaba and Wong, LAA 2012) Assume that 6 ≤ d. If λ2(G) < d −

5 d+1, then τ(G) ≥ 3.

– p. 11/39

Cioaba’s Problem

Let G be a d-regular graph. Theorem (Cioaba and Wong, LAA 2012) Assume that 4 ≤ d. If λ2(G) < d −

3 d+1, then τ(G) ≥ 2.

Theorem (Cioaba and Wong, LAA 2012) Assume that 6 ≤ d. If λ2(G) < d −

5 d+1, then τ(G) ≥ 3.

Conjecture (Cioaba and Wong, LAA 2012) Assume that 2 ≤ 2k ≤ d. If λ2(G) < d − 2k−1

d+1 , then τ(G) ≥ k.

– p. 11/39

Improvements in JGT, 2016

Can we work on generic graphs in stead of regular graphs?

– p. 12/39

Improvements in JGT, 2016

Can we work on generic graphs in stead of regular graphs? Let G be graph with δ(G) = δ and k > 0 be an integer.

– p. 12/39

Improvements in JGT, 2016

Can we work on generic graphs in stead of regular graphs? Let G be graph with δ(G) = δ and k > 0 be an integer. Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 4 and λ2(G) < δ −

3 δ+1, then τ(G) ≥ 2.

– p. 12/39

Improvements in JGT, 2016

Can we work on generic graphs in stead of regular graphs? Let G be graph with δ(G) = δ and k > 0 be an integer. Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 4 and λ2(G) < δ −

3 δ+1, then τ(G) ≥ 2.

Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 6 and λ2(G) < δ −

5 δ+1, then τ(G) ≥ 3.

– p. 12/39

Improvements in JGT, 2016

Let G be graph with δ(G) = δ and k > 0 be an integer.

– p. 13/39

Improvements in JGT, 2016

Let G be graph with δ(G) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d-regular, d ≥ 2k, and λ2(G) < d − 4k−2

d+1 , then τ(G) ≥ k.

– p. 13/39

Improvements in JGT, 2016

Let G be graph with δ(G) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d-regular, d ≥ 2k, and λ2(G) < d − 4k−2

d+1 , then τ(G) ≥ k.

Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 2k and λ2(G) < δ − 3k−1

δ+1 , then τ(G) ≥ k.

– p. 13/39

Improvements in JGT, 2016

Let G be graph with δ(G) = δ and k > 0 be an integer. Theorem (Cioaba, LAA 2010) If G is d-regular, d ≥ 2k, and λ2(G) < d − 4k−2

d+1 , then τ(G) ≥ k.

Theorem (X. Gu, P . Li, S. Yao and HJL, JGT 2016) If δ ≥ 2k and λ2(G) < δ − 3k−1

δ+1 , then τ(G) ≥ k.

Conjecture Let G be graph with δ(G) = δ, and 4 ≤ 2k ≤ δ. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

– p. 13/39

Over view of progresses

Conjecture (k, δ) Let G be graph with δ(G) = δ and 2k ≤ δ. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

– p. 14/39

Over view of progresses

Conjecture (k, δ) Let G be graph with δ(G) = δ and 2k ≤ δ. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

Let G be graph on n vertices with δ = δ(G) ≥ 2k ≥ 4.

– p. 14/39

Over view of progresses

Conjecture (k, δ) Let G be graph with δ(G) = δ and 2k ≤ δ. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

Let G be graph on n vertices with δ = δ(G) ≥ 2k ≥ 4. Theorem (G. Li and L. Shi, LAA 2013; Y. Hong, Q. Liu, and HJL, LAA 2014) For any integer k ≥ 2 and δ ≥ 2k, there exists an integer N = N(k, δ) such that if n ≥ N, then Conjecture(k, δ) holds,

– p. 14/39

Over view of progresses

Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2k ≥ 4. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

– p. 15/39

Over view of progresses

Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2k ≥ 4. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014)

– p. 15/39

Over view of progresses

Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2k ≥ 4. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014) How about Laplacian eigenvalues? (Algebraic connectivity)?

– p. 15/39

Over view of progresses

Conjecture (Gu et al.) Let G be a graph with minimum degree δ ≥ 2k ≥ 4. If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

It is a theorem. (Y. Hong, Q. Liu, Gu, and HJL, LAA 2014) How about Laplacian eigenvalues? (Algebraic connectivity)? How about signless Laplacian eigenvalues?

– p. 15/39

Over view of progresses

A = A(G): = adjacency matrix of G.

– p. 16/39

Over view of progresses

A = A(G): = adjacency matrix of G. D = D(G): = degree diagonal matrix of G.

– p. 16/39

Over view of progresses

A = A(G): = adjacency matrix of G. D = D(G): = degree diagonal matrix of G. A − D gives Laplacian eigenvalues.

– p. 16/39

Over view of progresses

A = A(G): = adjacency matrix of G. D = D(G): = degree diagonal matrix of G. A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues.

– p. 16/39

Over view of progresses

A = A(G): = adjacency matrix of G. D = D(G): = degree diagonal matrix of G. A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues. a: = a real number.

– p. 16/39

Over view of progresses

A = A(G): = adjacency matrix of G. D = D(G): = degree diagonal matrix of G. A − D gives Laplacian eigenvalues. D + A gives signless Laplacian eigenvalues. a: = a real number. λ1(G, a) ≥ λ2(G, a) ≥ · · · ≥ λn(G, a) are eigenvalues of aD + A.

– p. 16/39

Over view of progresses

λ1(G, a) ≥ λ2(G, a) ≥ · · · ≥ λn(G, a) are eigenvalues of aD + A.

– p. 17/39

Over view of progresses

λ1(G, a) ≥ λ2(G, a) ≥ · · · ≥ λn(G, a) are eigenvalues of aD + A.

• Theorem. (Liu, Hong, Gu, HJL, LAA 2014) Let k be an

integer and G be a graph of order n and minimum degree δ ≥ 2k. If λ2(G, a) < (a + 1)δ − 2k−1

δ+1 then τ(G) ≥ k.

– p. 17/39

Over view of progresses

λ1(G, a) ≥ λ2(G, a) ≥ · · · ≥ λn(G, a) are eigenvalues of aD + A.

• Theorem. (Liu, Hong, Gu, HJL, LAA 2014) Let k be an

integer and G be a graph of order n and minimum degree δ ≥ 2k. If λ2(G, a) < (a + 1)δ − 2k−1

δ+1 then τ(G) ≥ k.

Choose different values of a ∈ {0, 1, −1}.

– p. 17/39

Over view of progresses

λi(G): = the ith largest eigenvalue of A. µi(G): = the ith largest eigenvalue of D − A. qi(G): = the ith largest eigenvalue of D + A.

– p. 18/39

Over view of progresses

λi(G): = the ith largest eigenvalue of A. µi(G): = the ith largest eigenvalue of D − A. qi(G): = the ith largest eigenvalue of D + A.

• Theorem. (Liu, Hong, Gu, HJL, LAA 2014)

(1) If λ2(G) < δ − 2k−1

δ+1 , then τ(G) ≥ k.

(2) If q2(G) < 2δ − 2k−1

δ+1 , then τ(G) ≥ k.

(3) If µn−1(G) > 2k−1

δ+1 , then τ(G) ≥ k.

– p. 18/39

Outline of Proof of Cioaba-Wong Conjecture

The U-Lemma.

– p. 19/39

Outline of Proof of Cioaba-Wong Conjecture

The U-Lemma. Quadratic Inequality.

– p. 19/39

Outline of Proof of Cioaba-Wong Conjecture

The U-Lemma. Quadratic Inequality. Proof of Cioaba-Wong Conjecture.

– p. 19/39

Outline of Proof of Cioaba-Wong Conjecture

U-Lemma Let G be a graph with minimum degree δ > 0 and ∅ = U ⊂ V (G). If d(U) ≤ δ − 1, then |U| ≥ δ + 1.

– p. 20/39

Outline of Proof of Cioaba-Wong Conjecture

U-Lemma Let G be a graph with minimum degree δ > 0 and ∅ = U ⊂ V (G). If d(U) ≤ δ − 1, then |U| ≥ δ + 1. Proof: d(U) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [U, V − U].

– p. 20/39

Outline of Proof of Cioaba-Wong Conjecture

U-Lemma Let G be a graph with minimum degree δ > 0 and ∅ = U ⊂ V (G). If d(U) ≤ δ − 1, then |U| ≥ δ + 1. Proof: d(U) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [U, V − U]. NG(u) ⊆ U.

– p. 20/39

Outline of Proof of Cioaba-Wong Conjecture

U-Lemma Let G be a graph with minimum degree δ > 0 and ∅ = U ⊂ V (G). If d(U) ≤ δ − 1, then |U| ≥ δ + 1. Proof: d(U) ≤ δ − 1 means U has a vertex u ∈ U not incident with any edges in [U, V − U]. NG(u) ⊆ U. |U| ≥ |{u} ∪ NG(u)| ≥ 1 + δ.

– p. 20/39

Outline of Proof of Cioaba-Wong Conjecture

Lemma (Quadratic Inequality) Let X, Y ⊂ V (G) with X ∩ Y = ∅. If

– p. 21/39

Outline of Proof of Cioaba-Wong Conjecture

Lemma (Quadratic Inequality) Let X, Y ⊂ V (G) with X ∩ Y = ∅. If λ2(G, a) ≤ (a + 1)δ − max{ d(X)

|X| , d(Y ) |Y | }, then

– p. 21/39

Outline of Proof of Cioaba-Wong Conjecture

Lemma (Quadratic Inequality) Let X, Y ⊂ V (G) with X ∩ Y = ∅. If λ2(G, a) ≤ (a + 1)δ − max{ d(X)

|X| , d(Y ) |Y | }, then

|[X, Y ]|2 ≥ ((a + 1)δ − d(X) |X| − λ2(G, a)) · ((a + 1)δ − d(Y ) |Y | − λ2(G, a))|X| · |Y |.

– p. 21/39

Proof of Cioaba-Wong Conjecture (i)

Theorem Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2k. If λ2(G, a) < (a + 1)δ − 2k−1

δ+1

then τ(G) ≥ k.

– p. 22/39

Proof of Cioaba-Wong Conjecture (i)

Theorem Let k be an integer and G be a graph of order n and minimum degree δ ≥ 2k. If λ2(G, a) < (a + 1)δ − 2k−1

δ+1

then τ(G) ≥ k. Approach of the proof: For any partition (V1, V2, . . . , Vt), want to prove

1≤i<j≤t |[Vi, Vj]G| ≥ k(t − 1).

– p. 22/39

Proof of Cioaba-Wong Conjecture (ii)

Assume that d(V1) ≤ d(V2) ≤ . . . ≤ d(Vt).

– p. 23/39

Proof of Cioaba-Wong Conjecture (ii)

Assume that d(V1) ≤ d(V2) ≤ . . . ≤ d(Vt). If d(V1) ≥ 2k, then

1≤i<j≤t |[Vi, Vj]G| ≥ kt. Assume

d(V1) ≤ 2k − 1.

– p. 23/39

Proof of Cioaba-Wong Conjecture (ii)

Assume that d(V1) ≤ d(V2) ≤ . . . ≤ d(Vt). If d(V1) ≥ 2k, then

1≤i<j≤t |[Vi, Vj]G| ≥ kt. Assume

d(V1) ≤ 2k − 1. Let 1 ≤ s ≤ t be such that d(Vs) ≤ 2k − 1 and d(Vs+1) ≥ 2k (if s < t).

– p. 23/39

Proof of Cioaba-Wong Conjecture (ii)

Assume that d(V1) ≤ d(V2) ≤ . . . ≤ d(Vt). If d(V1) ≥ 2k, then

1≤i<j≤t |[Vi, Vj]G| ≥ kt. Assume

d(V1) ≤ 2k − 1. Let 1 ≤ s ≤ t be such that d(Vs) ≤ 2k − 1 and d(Vs+1) ≥ 2k (if s < t). By U-lemma, for 1 ≤ i ≤ s, |Vi| ≥ δ + 1.

– p. 23/39

Proof of Cioaba-Wong Conjectur (iii)

Assumption of Theorem, for 1 ≤ i ≤ s. λ2(G, a) < (a + 1)δ − 2k − 1 δ + 1 ≤ (a + 1)δ − d(Vi) |Vi| .

– p. 24/39

Proof of Cioaba-Wong Conjectur (iii)

Assumption of Theorem, for 1 ≤ i ≤ s. λ2(G, a) < (a + 1)δ − 2k − 1 δ + 1 ≤ (a + 1)δ − d(Vi) |Vi| . By Quadratic Inequality, for 2 ≤ i ≤ s, |[V1, Vi]|2 ≥

• (a + 1)δ − d(V1)

|V1| − λ2(G, a)

• ·
• (a + 1)δ − d(Vi)

|Vi| − λ2(G, a)

• |V1| · |Vi|

> (2k − 1 − d(V1))(2k − 1 − d(Vi)) ≥ (2k − 1 − d(Vi))2.

– p. 24/39

Proof of Cioaba-Wong Conjectur (iii)

Assumption of Theorem, for 1 ≤ i ≤ s. λ2(G, a) < (a + 1)δ − 2k − 1 δ + 1 ≤ (a + 1)δ − d(Vi) |Vi| . By Quadratic Inequality, for 2 ≤ i ≤ s, |[V1, Vi]|2 ≥

• (a + 1)δ − d(V1)

|V1| − λ2(G, a)

• ·
• (a + 1)δ − d(Vi)

|Vi| − λ2(G, a)

• |V1| · |Vi|

> (2k − 1 − d(V1))(2k − 1 − d(Vi)) ≥ (2k − 1 − d(Vi))2. |[V1, Vi]| > 2k − 1 − d(Vi), for 2 ≤ i ≤ s.

– p. 24/39

Proof of Cioaba-Wong Conjecture (iv)

Thus |[V1, Vi]| ≥ 2k − d(Vi), for 2 ≤ i ≤ s.

– p. 25/39

Proof of Cioaba-Wong Conjecture (iv)

Thus |[V1, Vi]| ≥ 2k − d(Vi), for 2 ≤ i ≤ s. d(V1) ≥ s

i=2 |[V1, Vi]| ≥ s i=2

• 2k − d(Vi)
• .

– p. 25/39

Proof of Cioaba-Wong Conjecture (iv)

Thus |[V1, Vi]| ≥ 2k − d(Vi), for 2 ≤ i ≤ s. d(V1) ≥ s

i=2 |[V1, Vi]| ≥ s i=2

• 2k − d(Vi)
• .

t

• i=1

d(Vi) = d(V1) +

s

• i=2

d(Vi) +

t

• i=s+1

d(Vi) ≥ 2k(s − 1) + 2k(t − s) = 2k(t − 1).

– p. 25/39

References

1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf).

– p. 26/39

References

1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040.

– p. 26/39

References

1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647.

– p. 26/39

References

1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647. 4 W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995), 593-616.

– p. 26/39

References

1 A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012. (http://homepages.cwi.nl/ aeb/math/ipm.pdf). 2 P . A. Catlin, H.-J. Lai and Y. Shao, Edge-connectivity and edge-disjoint spanning trees, Discrete Math., 309 (2009), 1033-1040. 3 S. M. Cioab˘ a and W.Wong, Edge-disjoint spanning trees and eigenvalues of regular graphs, Linear Algebra Appl., 437 (2012) 630-647. 4 W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995), 593-616. 5 G. Li and L. Shi, Edge-disjoint spanning trees and eigenvalues of graphs, Linear Algebra Appl. 439 (2013), 2784-2789.

– p. 26/39

References

6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29.

– p. 27/39

References

6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151.

– p. 27/39

References

6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133.

– p. 27/39

References

6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133. 9 Y. Hong, X. Gu, H. Lai, Q. Liu, Fractional spanning tree packing, forest covering and eigenvalues, Discrete Applied Math., 213 (2016) 219-223.

– p. 27/39

References

6 X. Gu, H. Lai, P . Li, S. Yao, Edge-disjoint spanning trees, edge connectivity and eigenvalues in graphs, J. Graph Theory, 81 (2016) 16-29. 7 Q. Liu, Y. Hong, H. Lai, Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 444 (2014) 146-151. 8 Q. Liu, Y. Hong, X. Gu, H. Lai, Note on Edge-disjoint spanning trees and eigenvalues, Linear Algebra Appl., 458 (2014), 128-133. 9 Y. Hong, X. Gu, H. Lai, Q. Liu, Fractional spanning tree packing, forest covering and eigenvalues, Discrete Applied Math., 213 (2016) 219-223.

– p. 27/39

Connectivity and eigenvalue

Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ2(G) to warrant κ(G) ≥ k.

– p. 28/39

Connectivity and eigenvalue

Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ2(G) to warrant κ(G) ≥ k. Let d and k be integers with d ≥ k ≥ 2 and G be a d-regular

• multigraph. Each of the following holds.

– p. 28/39

Connectivity and eigenvalue

Problem (Abiad, Brimkov, Mart´ lnez-Rivera, O, and Zhang, Electronic Journal of Linear Algebra, 2018) Find best possible condition on λ2(G) to warrant κ(G) ≥ k. Let d and k be integers with d ≥ k ≥ 2 and G be a d-regular

• multigraph. Each of the following holds.

Theorem (Suil O, arXiv:1603.03960v3 [math.CO] 4 Oct 2016.) If |V (G)| ≥ 3 and If |V (G)| ≥ 3 and λ2(G) < 3d

4 , then

κ(G) ≥ 2.

– p. 28/39

Connectivity and eigenvalue

Theorem (B. Brimkov, X. Mart´ lnez-Rivera, Suil O, J. Zhang, Electronic Journal of Linear Algebra, 2018). Suppose G is not spanned by a complete graph on at most k vertices, and

– p. 29/39

Connectivity and eigenvalue

Theorem (B. Brimkov, X. Mart´ lnez-Rivera, Suil O, J. Zhang, Electronic Journal of Linear Algebra, 2018). Suppose G is not spanned by a complete graph on at most k vertices, and let f(d, k) =              3 if G is a multigraph and k = 2; k if G is a multigraph and k ≥ 3; d + 2 if G is a simple graph and k = 2; d + 1 if G is a simple graph and k ≥ 3.

– p. 29/39

Connectivity and eigenvalue

Theorem (B. Brimkov, X. Mart´ lnez-Rivera, Suil O, J. Zhang, Electronic Journal of Linear Algebra, 2018). Suppose G is not spanned by a complete graph on at most k vertices, and let f(d, k) =              3 if G is a multigraph and k = 2; k if G is a multigraph and k ≥ 3; d + 2 if G is a simple graph and k = 2; d + 1 if G is a simple graph and k ≥ 3. If λ2(G) < d − (k−1)d

2f(d,k) − (k−1)d 2(n−f(d,k)) , then κ(G) ≥ k

– p. 29/39

Connectivity and eigenvalue

Theorem (B. Brimkov, X. Mart´ lnez-Rivera, Suil O, J. Zhang, Electronic Journal of Linear Algebra, 2018). Suppose G is not spanned by a complete graph on at most k vertices, and let f(d, k) =              3 if G is a multigraph and k = 2; k if G is a multigraph and k ≥ 3; d + 2 if G is a simple graph and k = 2; d + 1 if G is a simple graph and k ≥ 3. If λ2(G) < d − (k−1)d

2f(d,k) − (k−1)d 2(n−f(d,k)) , then κ(G) ≥ k

– p. 29/39

Connectivity and eigenvalue

Our goal: to study the relationship between connectivity and adjacency eigenvalues, algebraic connectivity (laplacian eigenvalues) and signless laplacian eigenvalues.

– p. 30/39

Connectivity and eigenvalue

Our goal: to study the relationship between connectivity and adjacency eigenvalues, algebraic connectivity (laplacian eigenvalues) and signless laplacian eigenvalues. We continuer using the matrix aD + A.

– p. 30/39

Connectivity and eigenvalue

Our goal: to study the relationship between connectivity and adjacency eigenvalues, algebraic connectivity (laplacian eigenvalues) and signless laplacian eigenvalues. We continuer using the matrix aD + A. λ1(G, a) ≥ λ2(G, a) ≥ · · · ≥ λn(G, a) are eigenvalues of aD + A.

– p. 30/39

Connectivity and eigenvalue

Given integers ∆, δ, k and g with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1, we have the following definition.

– p. 31/39

Connectivity and eigenvalue

Given integers ∆, δ, k and g with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1, we have the following definition. t = ⌊ g − 1 2 ⌋,

– p. 31/39

Connectivity and eigenvalue

Given integers ∆, δ, k and g with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1, we have the following definition. t = ⌊ g − 1 2 ⌋, ν = ν(δ, g, c) =              1 + (δ − c) t−1

i=0(δ − 1)i,

if g = 2t + 1 and c ≤ k − 1 ≤ δ − 2; 1 + 2(δ − 1)t−1 + t−2

i=0(δ − 1)i,

if g = 2t + 1 and c = k − 1 = δ − 1; 2 + (2δ − 2 − c) t−1

i=0(δ − 1)i,

if g = 2t + 2 and δ ≥ 3; 2t + 1, if g = 2t + 2 and δ = 2.

– p. 31/39

Connectivity and eigenvalue

Given integers ∆, δ, k and g with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1, we have the following definition. t = ⌊ g − 1 2 ⌋, ν = ν(δ, g, c) =              1 + (δ − c) t−1

i=0(δ − 1)i,

if g = 2t + 1 and c ≤ k − 1 ≤ δ − 2; 1 + 2(δ − 1)t−1 + t−2

i=0(δ − 1)i,

if g = 2t + 1 and c = k − 1 = δ − 1; 2 + (2δ − 2 − c) t−1

i=0(δ − 1)i,

if g = 2t + 2 and δ ≥ 3; 2t + 1, if g = 2t + 2 and δ = 2. Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) Each of the following holds. (i) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. (ii) If λ2(G, a) < (a + 1)δ − (k − 1)∆ ν(δ, g, k − 1) , then κ(G) ≥ k.

– p. 31/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k.

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

lim

n→∞

n n − ν = 1.

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

lim

n→∞

n n − ν = 1. For any ǫ > 0, there exists an integer N such that for any n ≥ N,

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

lim

n→∞

n n − ν = 1. For any ǫ > 0, there exists an integer N such that for any n ≥ N, 1 < n n − ν ≤ 1 + ǫ.

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

lim

n→∞

n n − ν = 1. For any ǫ > 0, there exists an integer N such that for any n ≥ N, 1 < n n − ν ≤ 1 + ǫ. λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν

– p. 32/39

Connectivity and eigenvalue

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) , then κ(G) ≥ k. Corollary For any ǫ > 0, there exists an integer N such that for any n ≥ N, if λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν(δ, g, k − 1) , then κ(G) ≥ k.

• Proof. Since ν = ν(δ, g, k − 1) > 0, and since

lim

n→∞

n n − ν = 1. For any ǫ > 0, there exists an integer N such that for any n ≥ N, 1 < n n − ν ≤ 1 + ǫ. λ2(G, a) < (a + 1)δ − (k − 1)∆(1 + ǫ) 2ν ≤ (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)) .

– p. 32/39

Connectivity and eigenvalue

Let ∆, δ, k and g be integers with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1.

– p. 33/39

Connectivity and eigenvalue

Let ∆, δ, k and g be integers with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1. Define α = ⌈ δ+1+√

(δ+1)2−2(k−1)∆ 2

⌉, and

– p. 33/39

Connectivity and eigenvalue

Let ∆, δ, k and g be integers with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1. Define α = ⌈ δ+1+√

(δ+1)2−2(k−1)∆ 2

⌉, and φ = φ(δ, ∆, k) =    (δ − k + 2)(n − δ + k − 2), if ∆ ≥ 2(δ − k + 2); α(n − α), if δ ≤ ∆ < 2(δ − k + 2).

– p. 33/39

Connectivity and eigenvalue

Let ∆, δ, k and g be integers with ∆ ≥ δ ≥ k ≥ 2, g ≥ 3 and 1 ≤ c ≤ k − 1. Define α = ⌈ δ+1+√

(δ+1)2−2(k−1)∆ 2

⌉, and φ = φ(δ, ∆, k) =    (δ − k + 2)(n − δ + k − 2), if ∆ ≥ 2(δ − k + 2); α(n − α), if δ ≤ ∆ < 2(δ − k + 2). Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) Each of the following holds. (i) If λ2(G) < δ − (k−1)∆n

2φ(δ,∆,k) , then κ(G) ≥ k.

(ii) If µn−1(G) > (k−1)∆n

2φ(δ,∆,k) , then κ(G) ≥ k.

(iii) If q2(G) < 2δ − (k−1)∆n

2φ(δ,∆,k) , then κ(G) ≥ k.

– p. 33/39

Outline of Proof

We need to modify the Useful Lemma.

– p. 34/39

Outline of Proof

We need to modify the Useful Lemma. t = ⌊ g − 1 2 ⌋,

– p. 34/39

Outline of Proof

We need to modify the Useful Lemma. t = ⌊ g − 1 2 ⌋, ν = ν(δ, g, c) =              1 + (δ − c) t−1

i=0(δ − 1)i,

if g = 2t + 1 and c ≤ k − 1 ≤ δ − 2; 1 + 2(δ − 1)t−1 + t−2

i=0(δ − 1)i,

if g = 2t + 1 and c = k − 1 = δ − 1; 2 + (2δ − 2 − c) t−1

i=0(δ − 1)i,

if g = 2t + 2 and δ ≥ 3; 2t + 1, if g = 2t + 2 and δ = 2.

– p. 34/39

Outline of Proof

We need to modify the Useful Lemma. t = ⌊ g − 1 2 ⌋, ν = ν(δ, g, c) =              1 + (δ − c) t−1

i=0(δ − 1)i,

if g = 2t + 1 and c ≤ k − 1 ≤ δ − 2; 1 + 2(δ − 1)t−1 + t−2

i=0(δ − 1)i,

if g = 2t + 1 and c = k − 1 = δ − 1; 2 + (2δ − 2 − c) t−1

i=0(δ − 1)i,

if g = 2t + 2 and δ ≥ 3; 2t + 1, if g = 2t + 2 and δ = 2. New Useful Lemma. Let G be a simple connected graph with δ = δ(G) ≥ k ≥ 2 and girth g = g(G) ≥ 3. Let C be a minimum vertex cut of G with |C| = c and U be a connected component of G − C. If c ≤ k − 1 < δ, then |V (U)| ≥ ν(δ, g, c).

– p. 34/39

Outline of Proof

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) Each

• f the following holds.

(i) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)), then κ(G) ≥ k. (ii) If λ2(G, a) < (a + 1)δ − (k − 1)∆ ν(δ, g, k − 1), then κ(G) ≥ k.

– p. 35/39

Outline of Proof

Theorem (R. Liu, Y. Tian, Y. Wu and HJL, AMC 2019) Each

• f the following holds.

(i) If λ2(G, a) < (a + 1)δ − (k − 1)∆n 2ν(δ, g, k − 1)(n − ν(δ, g, k − 1)), then κ(G) ≥ k. (ii) If λ2(G, a) < (a + 1)δ − (k − 1)∆ ν(δ, g, k − 1), then κ(G) ≥ k. By contradiction, we assume κ(G) = c ≤ k − 1.

– p. 35/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1.

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y .

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y . A:= a connected component of G − C, m1 = e(A, C) = d(A).

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y . A:= a connected component of G − C, m1 = e(A, C) = d(A). B := G − (V (A) ∪ C), m2 = e(B, C), and A = V (G) − A.

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y . A:= a connected component of G − C, m1 = e(A, C) = d(A). B := G − (V (A) ∪ C), m2 = e(B, C), and A = V (G) − A. ν = ν(δ, g, k − 1), |A| = n1 and |B| = n2.

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y . A:= a connected component of G − C, m1 = e(A, C) = d(A). B := G − (V (A) ∪ C), m2 = e(B, C), and A = V (G) − A. ν = ν(δ, g, k − 1), |A| = n1 and |B| = n2. By New Useful Lemma, ν ≤ min{n1, n2} ≤ n

2 ≤ n − ν.

– p. 36/39

Outline of Proof

C:= a minimum vertex cut of G, |C| = c ≤ k − 1 ≤ δ − 1. Notation: For subsets X, Y ⊂ V (G), e(X, Y ):= number of edges in G linking a vertex in X and a vertex in Y . A:= a connected component of G − C, m1 = e(A, C) = d(A). B := G − (V (A) ∪ C), m2 = e(B, C), and A = V (G) − A. ν = ν(δ, g, k − 1), |A| = n1 and |B| = n2. By New Useful Lemma, ν ≤ min{n1, n2} ≤ n

2 ≤ n − ν.

n ≥ 2ν or n 2(n − ν) ≤ 1.

– p. 36/39

Outline of Proof

Let ¯ d1 = 1 n1

• v∈A

dG(v) and ¯ d2 = 1 n2 + c

• v∈A

dG(v).

– p. 37/39

Outline of Proof

Let ¯ d1 = 1 n1

• v∈A

dG(v) and ¯ d2 = 1 n2 + c

• v∈A

dG(v). he quotient matrix of aD + A corresponding to the partition (A, C ∪ B) becomes:

– p. 37/39

Outline of Proof

Let ¯ d1 = 1 n1

• v∈A

dG(v) and ¯ d2 = 1 n2 + c

• v∈A

dG(v). he quotient matrix of aD + A corresponding to the partition (A, C ∪ B) becomes: R(aD + A) =   (a + 1) ¯ d1 − m1

n1 m1 n1 m1 n2+c

(a + 1) ¯ d2 −

m1 n2+c

  .

– p. 37/39

Outline of Proof

Let ¯ d1 = 1 n1

• v∈A

dG(v) and ¯ d2 = 1 n2 + c

• v∈A

dG(v). he quotient matrix of aD + A corresponding to the partition (A, C ∪ B) becomes: R(aD + A) =   (a + 1) ¯ d1 − m1

n1 m1 n1 m1 n2+c

(a + 1) ¯ d2 −

m1 n2+c

  . Apply n ≥ 2ν or n 2(n − ν) ≤ 1 and algebra,

– p. 37/39

Outline of Proof

Let ¯ d1 = 1 n1

• v∈A

dG(v) and ¯ d2 = 1 n2 + c

• v∈A

dG(v). he quotient matrix of aD + A corresponding to the partition (A, C ∪ B) becomes: R(aD + A) =   (a + 1) ¯ d1 − m1

n1 m1 n1 m1 n2+c

(a + 1) ¯ d2 −

m1 n2+c

  . Apply n ≥ 2ν or n 2(n − ν) ≤ 1 and algebra, to conclude λ2(R(aD + A)) ≥ (a + 1)δ − (k − 1)∆n 2ν(n − ν) .

– p. 37/39

Outline of Proof

Given two sequences θ1 ≥ θ2 ≥ · · · θn and η1 ≥ η2 ≥ · · · ≥ ηm with n > m, the second sequence interlaces the first if θi ≥ ηi ≥ θn−m+i, for 1 ≤ i ≤ m.

– p. 38/39

Outline of Proof

Given two sequences θ1 ≥ θ2 ≥ · · · θn and η1 ≥ η2 ≥ · · · ≥ ηm with n > m, the second sequence interlaces the first if θi ≥ ηi ≥ θn−m+i, for 1 ≤ i ≤ m. Theorem (Haemers, LAA 1995) Eigenvalues of any quotient matrix of G interlace the eigenvalues of G.

– p. 38/39

Outline of Proof

Given two sequences θ1 ≥ θ2 ≥ · · · θn and η1 ≥ η2 ≥ · · · ≥ ηm with n > m, the second sequence interlaces the first if θi ≥ ηi ≥ θn−m+i, for 1 ≤ i ≤ m. Theorem (Haemers, LAA 1995) Eigenvalues of any quotient matrix of G interlace the eigenvalues of G. By interlacing (we have a contradiction) λ2(aD + A) ≥ λ2(R(aD + A)) ≥ (a + 1)δ − (k − 1)∆n 2ν(n − ν) .

– p. 38/39

Thank You

– p. 39/39