OPERATIONS ON GRAPHS INCREASING SOME GRAPH PARAMETERS Alexander - - PowerPoint PPT Presentation

OPERATIONS ON GRAPHS INCREASING SOME GRAPH PARAMETERS Alexander Kelmans

University of Puerto Rico Rutgers University May 25, 2014

• 1. Let Gm

n be the set of graphs with n vertices and m edges.

Let Q be an operation on a graph such that G ∈ Gm

n

⇒ Q(G) ∈ Gm

n .

• 1. Let Gm

n be the set of graphs with n vertices and m edges.

Let Q be an operation on a graph such that G ∈ Gm

n

⇒ Q(G) ∈ Gm

n .

• 2. Let (Gm

n , ) be a quasi-poset. An operation Q is called

• increasing (-decreasing) if

Q(G) G (resp., Q(G) G) for every G ∈ Gm

n .

• 1. A graph G is called vertex comparable (A.K. 1970) if

N(x, G) \ x ⊆ N(y, G) \ y or N(y, G) \ y ⊆ N(x, G) \ x for every x, y ∈ V (G).

• 1. A graph G is called vertex comparable (A.K. 1970) if

N(x, G) \ x ⊆ N(y, G) \ y or N(y, G) \ y ⊆ N(x, G) \ x for every x, y ∈ V (G).

• 2. A graph G is called threshold (V. Chv´

atal, P. Hammer 1973) if G has no induced , N or II.

• 1. A graph G is called vertex comparable (A.K. 1970) if

N(x, G) \ x ⊆ N(y, G) \ y or N(y, G) \ y ⊆ N(x, G) \ x for every x, y ∈ V (G).

• 2. A graph G is called threshold (V. Chv´

atal, P. Hammer 1973) if G has no induced , N or II.

• 3. Claim. G is vertex comparable if and only if G is threshold.

• 1. Let k, r, s be integers, k ≥ 0, and 0 ≤ r < s. Let F(k, r, s) be

the graph obtained from the complete graph Ks as follows:

• fix in Ks a set A of r vertices and a ∈ A,
• add to Ks a new vertex c and the set {cx : x ∈ A} of new

edges to obtain graph C(r, s), and

• add to C(r, s) the set B of k new vertices and the set

{az : z ∈ B} of new edge to obtain graph F(k, r, s).

• 1. Let k, r, s be integers, k ≥ 0, and 0 ≤ r < s. Let F(k, r, s) be

the graph obtained from the complete graph Ks as follows:

• fix in Ks a set A of r vertices and a ∈ A,
• add to Ks a new vertex c and the set {cx : x ∈ A} of new

edges to obtain graph C(r, s), and

• add to C(r, s) the set B of k new vertices and the set

{az : z ∈ B} of new edge to obtain graph F(k, r, s).

• 2. Let Cm

n be the set of connected graphs with n vertices and m

edges. Claim. For every pair (n, m) of integers such that Cm

n = ∅

there exists a unique triple (k, r, s) of integers such that k ≥ 0, 0 ≤ r < s, and F(k, r, s) ∈ Cm

n .

We call F(k, r, s) = F m

n the extreme graph in Cm n .

• 1. Theorem (A.K. 1970). Let n and m be natural numbers and

n ≥ 3. (a1) If n − 1 ≤ m ≤ 2n − 4, then F m

n is the only threshold graph

with n vertices and m edges, i.e. Fm

n = {F m n }.

(a2) If m = 2n − 3, then F m

n is not the only threshold graph

with n vertices and m edges.

• 1. Theorem (A.K. 1970). Let n and m be natural numbers and

n ≥ 3. (a1) If n − 1 ≤ m ≤ 2n − 4, then F m

n is the only threshold graph

with n vertices and m edges, i.e. Fm

n = {F m n }.

(a2) If m = 2n − 3, then F m

n is not the only threshold graph

with n vertices and m edges.

• 2. Theorem (A.K. 1970). Let G be a connected graph. Then

(a1) there exists a connected threshold graph F obtained from G by a series of ♦-operations, and so (a2) if the ♦-operation is -decreasing, then there exists a connected threshold graph F such that G F.

• 1. Theorem (A.K. 1970). Let the ♦-operation be -decreasing,

F a graph with r edges and at most n vertices, and rP1 and Sr are a matching and a star with r edges. Then Kn−E(rP1) Kn−E(S2+(r−2)P1) Kn−E(F) Kn−E(Sr), where r ≥ 2, r ≤ n/2 for the second , and r ≤ n − 1 for the last .

• 1. Theorem (A.K. 1970). Let the ♦-operation be -decreasing,

F a graph with r edges and at most n vertices, and rP1 and Sr are a matching and a star with r edges. Then Kn−E(rP1) Kn−E(S2+(r−2)P1) Kn−E(F) Kn−E(Sr), where r ≥ 2, r ≤ n/2 for the second , and r ≤ n − 1 for the last .

• 2. Theorem (A.K. 1970). Let the ♦-operation be -decreasing,

G ∈ Cm

n , and ¨

G be obtained from G by adding m − n + 1 isolated vertices. Then for every spanning tree T of G there exists a tree D with m edges such that T is a subgraph of D and D ¨ G.

Theorem (A.K. 1970) Let G ∈ Cm

n , Pn an n-vertex path, Cn an n-vertex cycle, and

G ∈ F m

n . Suppose that the ♦-operation is -decreasing.

(a1) If m = n − 1 ≥ 3 and G = Pn, then Pn ≻ G ≻ F n−1

n

. (a2) If m = n ≥ 3 and G = Cn, then Cn ≻ G ≻ F n

n .

(a3) If n ≥ 4 and m = n + 1, then G ≻ F n+1

n

. (a4) If n ≥ 5 and n + 2 ≤ m ≤ 2n − 4, then G ≻ F m

n .

• 1. Suppose that every edge of a graph G has probability p to exist

and the edge events are independent.

• 1. Suppose that every edge of a graph G has probability p to exist

and the edge events are independent.

• 2. Let R(p, G) be the probability that the random graph (G, p)

is connected. We call R(p, G) the the reliability of G.

• 1. Suppose that every edge of a graph G has probability p to exist

and the edge events are independent.

• 2. Let R(p, G) be the probability that the random graph (G, p)

is connected. We call R(p, G) the the reliability of G.

• 3. Then

R(p, G) =

• { as(G) ps qm−s : s ∈ {n − 1, . . . , m} },

where n and m are the numbers of vertices and edges of G, q = 1 − p, and as(G) is the number of connected spanning subgraphs of G with s edges, and so an−1 = t(G) is the number of spanning trees of G.

• 1. Problem

Find a most reliable graph M(p) with n vertices and m edges, i.e. such that R(p, M(p)) = max { R(p, G) : G ∈ Gm

n }.

• 1. Problem

Find a most reliable graph M(p) with n vertices and m edges, i.e. such that R(p, M(p)) = max { R(p, G) : G ∈ Gm

n }.

• 2. Problem

Find a least reliable connected graph L(p) with n vertices and m edges, i.e. such that R(p, L(p)) = min { R(p, G) : G ∈ Cm

n }.

• 1. Problem

Find a most reliable graph M(p) with n vertices and m edges, i.e. such that R(p, M(p)) = max { R(p, G) : G ∈ Gm

n }.

• 2. Problem

Find a least reliable connected graph L(p) with n vertices and m edges, i.e. such that R(p, L(p)) = min { R(p, G) : G ∈ Cm

n }.

• 3. Problem

Find a graph Am

n ∈ Gm n with the maximum number

as(G) of connected spanning subgraps with s edges: as(F m

n ) = max { as(G) : G ∈ Gm n }.

• 1. Problem

Find a most reliable graph M(p) with n vertices and m edges, i.e. such that R(p, M(p)) = max { R(p, G) : G ∈ Gm

n }.

• 2. Problem

Find a least reliable connected graph L(p) with n vertices and m edges, i.e. such that R(p, L(p)) = min { R(p, G) : G ∈ Cm

n }.

• 3. Problem

Find a graph Am

n ∈ Gm n with the maximum number

as(G) of connected spanning subgraps with s edges: as(F m

n ) = max { as(G) : G ∈ Gm n }.

• 4. Problem

Find a graph Bm

n

∈ Gm

n with the maximum number

• f spanning trees, i.e. such that

t(Bm

n ) = max { t(G) : G ∈ Gm n }.

• 1. Poset (Gm

n , r):

G r F if R(p, G) ≥ R(p, F) for every p ∈ [0, 1].

• 1. Poset (Gm

n , r):

G r F if R(p, G) ≥ R(p, F) for every p ∈ [0, 1].

• 2. Poset (Gm

n , a):

G a F if as(G) ≥ as(F) for s ∈ {n − 1, . . . , m}.

• 1. Poset (Gm

n , r):

G r F if R(p, G) ≥ R(p, F) for every p ∈ [0, 1].

• 2. Poset (Gm

n , a):

G a F if as(G) ≥ as(F) for s ∈ {n − 1, . . . , m}.

• 3. Poset (Gm

n , t):

G t F if t(G) ≥ t(F).

• 1. Poset (Gm

n , r):

G r F if R(p, G) ≥ R(p, F) for every p ∈ [0, 1].

• 2. Poset (Gm

n , a):

G a F if as(G) ≥ as(F) for s ∈ {n − 1, . . . , m}.

• 3. Poset (Gm

n , t):

G t F if t(G) ≥ t(F).

• 4. Obviously,

a ⇒ r ⇒ t .

• 1. Poset (Gm

n , r):

G r F if R(p, G) ≥ R(p, F) for every p ∈ [0, 1].

• 2. Poset (Gm

n , a):

G a F if as(G) ≥ as(F) for s ∈ {n − 1, . . . , m}.

• 3. Poset (Gm

n , t):

G t F if t(G) ≥ t(F).

• 4. Obviously,

a ⇒ r ⇒ t .

• 5. Theorem (A.K. 1966)

Let ∈ {a, r, t}. Let G, G ′ ∈ Gm

n and G ′ = Hxy(G). Then G G ′.

• 1. Let L(λ, G) be the characteristic polynomial of the Laplacian

matrix of G.

• 1. Let L(λ, G) be the characteristic polynomial of the Laplacian

matrix of G.

• 2. Poset (Gm

n , L):

G L F if L(λ, G) ≥ L(λ, F) for λ ≥ n.

• 1. Let L(λ, G) be the characteristic polynomial of the Laplacian

matrix of G.

• 2. Poset (Gm

n , L):

G L F if L(λ, G) ≥ L(λ, F) for λ ≥ n.

• 3. Poset (Gm

n , τ)):

G τ F if t(Kn+r − E(G)) ≥ t(Kn+r − E(F)) for integer r ≥ 0.

• 1. Let L(λ, G) be the characteristic polynomial of the Laplacian

matrix of G.

• 2. Poset (Gm

n , L):

G L F if L(λ, G) ≥ L(λ, F) for λ ≥ n.

• 3. Poset (Gm

n , τ)):

G τ F if t(Kn+r − E(G)) ≥ t(Kn+r − E(F)) for integer r ≥ 0.

• 4. Claim (A.K. 1965). L

⇒ τ .

• 1. Let L(λ, G) be the characteristic polynomial of the Laplacian

matrix of G.

• 2. Poset (Gm

n , L):

G L F if L(λ, G) ≥ L(λ, F) for λ ≥ n.

• 3. Poset (Gm

n , τ)):

G τ F if t(Kn+r − E(G)) ≥ t(Kn+r − E(F)) for integer r ≥ 0.

• 4. Claim (A.K. 1965). L

⇒ τ .

• 5. Theorem (A.K. 1966)

Let G, G ′ ∈ Gm

n and G ′ = Hxy(G).

Then G L G ′, and so G τ G ′.

• 1. Theorem (A.K. 1970)

Let F be a forest, Cmp(F) the set of components of F, F(G) the set of spanning forests of G, and γ(F) = { v(C) : C ∈ Cmp(F) }. Then L(λ, G) = { (−1)s cs(G) λn−s : s ∈ {0, . . . , n − 1} }, where cs(G) = { γ(F) : F ∈ F(G), e(F) = s }.

• 1. Theorem (A.K. 1970)

Let F be a forest, Cmp(F) the set of components of F, F(G) the set of spanning forests of G, and γ(F) = { v(C) : C ∈ Cmp(F) }. Then L(λ, G) = { (−1)s cs(G) λn−s : s ∈ {0, . . . , n − 1} }, where cs(G) = { γ(F) : F ∈ F(G), e(F) = s }.

• 2. Poset (Gm

n , c):

G c F if cs(G) ≥ cs(F) for every s ∈ {0, . . . , n − 1}.

• 1. Theorem (A.K. 1970)

Let F be a forest, Cmp(F) the set of components of F, F(G) the set of spanning forests of G, and γ(F) = { v(C) : C ∈ Cmp(F) }. Then L(λ, G) = { (−1)s cs(G) λn−s : s ∈ {0, . . . , n − 1} }, where cs(G) = { γ(F) : F ∈ F(G), e(F) = s }.

• 2. Poset (Gm

n , c):

G c F if cs(G) ≥ cs(F) for every s ∈ {0, . . . , n − 1}.

• 3. Theorem (A.K. 1995) Let G, G ′ ∈ Gm

n

and G ′ = Hxy(G). Then G L,c G ′.

• 1. Given a symmetric function σ on k variables and a graph F with

k components, let σ[F] = σ{ v(C) : C ∈ Cmp(F) }. For a graph G with n vertices, let ˜ cs(G) =

• { σ[F] : F ∈ F(G), e(F) = s },

where σ is a symmetric function of n − s variables.

• 1. Given a symmetric function σ on k variables and a graph F with

k components, let σ[F] = σ{ v(C) : C ∈ Cmp(F) }. For a graph G with n vertices, let ˜ cs(G) =

• { σ[F] : F ∈ F(G), e(F) = s },

where σ is a symmetric function of n − s variables.

• 2. Theorem (A.K. 1995)

Let G, G ′ ∈ Gm

n

and G ′ = Hxy(G). Suppose that σ is a symmetric concave function. Then ˜ cs(G) ≥ ˜ cs(G ′).

• 1. Let A(λ, G) be the characteristic polynomial of the adjacency

matrix of G and α(G) the maximum eigenvalue of A(G).

• 1. Let A(λ, G) be the characteristic polynomial of the adjacency

matrix of G and α(G) the maximum eigenvalue of A(G).

• 2. Poset (Gm

n , A):

G A F if A(λ, G) ≥ A(λ, F) for every λ ≥ α(F).

• 1. Let A(λ, G) be the characteristic polynomial of the adjacency

matrix of G and α(G) the maximum eigenvalue of A(G).

• 2. Poset (Gm

n , A):

G A F if A(λ, G) ≥ A(λ, F) for every λ ≥ α(F).

• 3. Theorem (A.K. 1992)

Let G, G ′ ∈ Gm

n

and G ′ = Hxy(G). Then G A G ′.

• 1. Poset (Gm

n , h):

G h F if hi(G) ≥ hi(F) for i ∈ {0, 1}, where h0(G) and h1(G) are the numbers of Hamiltonian cycles and paths in G.

• 1. Poset (Gm

n , h):

G h F if hi(G) ≥ hi(F) for i ∈ {0, 1}, where h0(G) and h1(G) are the numbers of Hamiltonian cycles and paths in G.

• 2. Theorem (A.K. 1970)

Let G, G ′ ∈ Gm

n and G ′ = Hxy(G). Then

G h G ′.

Theorem (A.K. 1995) Let the graph Gb be obtained from a graph Ga by the operation on the above figure. Suppose that (h1) the two-pole xHy is symmetric and (h2) F has a path bBt such that v(A) ≤ v(B). Then Ga c Gb and v(A) < v(B) ⇒ Ga ≻≻c Gb.

• 1. Let

Φ(λ, G) = λm−n L(λ, G) and λ(G) the maximum Laplacian eigenvalue of G.

• 1. Let

Φ(λ, G) = λm−n L(λ, G) and λ(G) the maximum Laplacian eigenvalue of G.

• 2. Poset (Gm

n , φ):

G φ F if λ(G) ≤ λ(F) and Φ(λ, G) ≥ Φ(λ, F) for every λ ≥ λ(F).

• 1. Let

Φ(λ, G) = λm−n L(λ, G) and λ(G) the maximum Laplacian eigenvalue of G.

• 2. Poset (Gm

n , φ):

G φ F if λ(G) ≤ λ(F) and Φ(λ, G) ≥ Φ(λ, F) for every λ ≥ λ(F).

• 3. Clearly, φ

⇒ L.

• 1. Let

Φ(λ, G) = λm−n L(λ, G) and λ(G) the maximum Laplacian eigenvalue of G.

• 2. Poset (Gm

n , φ):

G φ F if λ(G) ≤ λ(F) and Φ(λ, G) ≥ Φ(λ, F) for every λ ≥ λ(F).

• 3. Clearly, φ

⇒ L.

• 4. Theorem (A.K. 1970)

Let the graph Gb be obtained from a graph Ga by the operation

• n the above figure. Then Ga φ Gb.

≻: = ≻p

• 1. Let Dn(r) and Kn(r) denote the sets of n-vertex trees and

caterpillars of diameter r, and so Kn(r) ⊆ Dn(r).

• 1. Let Dn(r) and Kn(r) denote the sets of n-vertex trees and

caterpillars of diameter r, and so Kn(r) ⊆ Dn(r).

• 2. Let Kn(r) be the n-vertex graph obtained from a disjoint path

P with r ≥ 2 edges and a star S by identifying a center vertex

• f P and a center of S, and so Kn(r) ∈ Kn(r).

K = Kn(r), where v(K) = n and diam(K) = r

• 1. Theorem (A.K. 1970 and 1995, resp.)

Let r ≥ 3 and n ≥ r + 2. Then (a1) (Dn(3), φ,c) and (Dn(4), φ,c) are linear posets, (a2) for every D ∈ Dn(r) \ Kn(r) there exists Y ∈ Kn(r) such that D ≻≻φ,c Y , (a3) D ≻≻φ,c Kn(r) for every D ∈ Kn(r) \ {Kn(r)}, and therefore (from (a1) and (a2)) (a4) D ≻≻φ,c Kn(r) for every D ∈ Dn(r) \ {Kn(r)}.

• 1. Let Ln(r) denote the sets of n-vertex trees having r leaves.

• 1. Let Ln(r) denote the sets of n-vertex trees having r leaves.
• 2. Let Sn(r), r ≥ 3, be the set of n-vertex trees T such that T has

exactly one vertex of degree r and every other vertex in T has degree at most two, and so Sn(r) ⊆ Ln(r).

• 1. Let Ln(r) denote the sets of n-vertex trees having r leaves.
• 2. Let Sn(r), r ≥ 3, be the set of n-vertex trees T such that T has

exactly one vertex of degree r and every other vertex in T has degree at most two, and so Sn(r) ⊆ Ln(r).

• 3. Let Mn(r) be the tree T in S(r) obtained from a disjoint path P

and a star S by identifying an end-vertex of P with the center

• f S.

M = Mn(r), where v(M) = n and lv(M) = r

Let Ln(r) be the tree T in Sn(r) such that |e(P) − e(Q)| ≤ 1 for every two components P and Q of T − z, where z is the vertex

• f degree r in T.

L = Ln(r), where v(L) = n and lv(L) = r

Theorem (A.K. 1970 and 1995, resp.) Let r ≥ 3 and n ≥ r + 2. Then (a0) Ln(r) ≻≻φ,c Ln(r + 1) for every r ∈ {2, . . . , n − 2}, (a1) (Sn(r), φ,c) is a linear poset, (a2) Mn(r) ≻≻φ,c L for every L ∈ Sn(r) \ {Mn(r)}, (a3) for every L ∈ Ln(r) \ Sn(r) there exists Z ∈ Sn(r) such that L ≻≻φ,c Z, (a4) L ≻≻φ,c Ln(r) for every L ∈ Sn(r) \ {Ln(r)}, and therefore (a5) L ≻≻φ,c Ln(r) for every L ∈ Ln(r) \ {Ln(r)}, (a6) λ(Ln(r)) > λ(L) for every L ∈ Ln(r) \ {Ln(r)}, and (a7) If T is an n-vertex tree with the maximum degree r and T is not isomorphic to Mn(r), then Mn(r) ≻≻φ,c T.

Theorem (A.K. 1995) Let G ∈ Cm

n , Pn an n-vertex path,

Cn an n-vertex cycle, and G = F m

n .

(a1) If m = n − 1 ≥ 3 and G = Pn, then Pn ≻L G ≻L F n−1

n

, cs(Pn) > cs(G) > cs(F n−1

n

) for every s ∈ {2, . . . , n − 2}, and cn−1(G) = cn−1(F n−1

n

) = n. (a2) If m = n ≥ 3 and G = Cn, then Cn ≻L G ≻L F n

n ,

cs(Cn) > cs(G) > cs(F n

n ) for every s ∈ {2, . . . , n − 2}, and

cn−1(G) ≥ cn−1(F n

n ).

(a3) If n ≥ 4 and m = n + 1, then G ≻L F n+1

n

, cs(G) > cs(F n+1

n

) for every s ∈ {2, . . . , n − 2}, and cn−1(G) ≥ cn−1(F n+1

n

).

Theorem (A.K. 1995) Let G ∈ Cm

n and G = F m n .

(a1) If n ≥ 5 and n + 2 ≤ m ≤ 2n − 4, then G ≻L F m

n , cs(G) > cs(F m n ) for every s ∈ {2, . . . , n − 2}, and

cn−1(G) = cn−1(F m

n ).

(a2) If m = 2n − 3, then for every n ≥ 6 there exists G ∈ Cm

n

such that G c F m

n .

• 1. Let M(x, G) be the matching polynomial of a graph G:

M(x, G) =

• { (−1)r µr(G) xn−2r : r ∈ {0, . . . , ⌊n/2⌋} },

where µr(G) is the number of r-matchings in G.

• 1. Let M(x, G) be the matching polynomial of a graph G:

M(x, G) =

• { (−1)r µr(G) xn−2r : r ∈ {0, . . . , ⌊n/2⌋} },

where µr(G) is the number of r-matchings in G.

• 2. Claim.

The roots of M(x, G) are real numbers. Let ρ(G) be the largest root of M(x, G).

• 1. Let M(x, G) be the matching polynomial of a graph G:

M(x, G) =

• { (−1)r µr(G) xn−2r : r ∈ {0, . . . , ⌊n/2⌋} },

where µr(G) is the number of r-matchings in G.

• 2. Claim.

The roots of M(x, G) are real numbers. Let ρ(G) be the largest root of M(x, G).

• 3. Poset (Gm

n , M):

G M F if M(x, G) ≥ M(x, F) for every x ≥ ρ(F).

• 1. Let M(x, G) be the matching polynomial of a graph G:

M(x, G) =

• { (−1)r µr(G) xn−2r : r ∈ {0, . . . , ⌊n/2⌋} },

where µr(G) is the number of r-matchings in G.

• 2. Claim.

The roots of M(x, G) are real numbers. Let ρ(G) be the largest root of M(x, G).

• 3. Poset (Gm

n , M):

G M F if M(x, G) ≥ M(x, F) for every x ≥ ρ(F).

• 4. Poset (Gm

n , µ):

G µ F if µr(G) ≥ µr(F) for every r ∈ {0, . . . , ⌊n/2⌋}.

• 1. Let I(x, G) be the independence polynomial of a graph G:

I(x, G) =

• { (−1)s is(G) xs : s ∈ {0, . . . , n} },

where is(G) is the number of independent sets of size s in G.

• 1. Let I(x, G) be the independence polynomial of a graph G:

I(x, G) =

• { (−1)s is(G) xs : s ∈ {0, . . . , n} },

where is(G) is the number of independent sets of size s in G.

• 2. Claim.

I(x, G) has a real root and every real root is positive. Let r(G) be the smallest real root of I(x, G).

• 1. Let I(x, G) be the independence polynomial of a graph G:

I(x, G) =

• { (−1)s is(G) xs : s ∈ {0, . . . , n} },

where is(G) is the number of independent sets of size s in G.

• 2. Claim.

I(x, G) has a real root and every real root is positive. Let r(G) be the smallest real root of I(x, G).

• 3. Poset (Gm

n , I):

G I F if I(x, G) ≥ I(x, F) for every x ∈ [0, r(F)].

• 1. Let λ be a positive integer and X(λ, G) be the number of proper

colorings of G with λ colors.

• 1. Let λ be a positive integer and X(λ, G) be the number of proper

colorings of G with λ colors.

• 2. Claim.

X(λ, G) is a polynomial (called the chromatic polynomial of a graph G): X(λ, G) =

• { (−1)n−i χi(G) λi : i ∈ {1, . . . , n} }.

• 1. Let λ be a positive integer and X(λ, G) be the number of proper

colorings of G with λ colors.

• 2. Claim.

X(λ, G) is a polynomial (called the chromatic polynomial of a graph G): X(λ, G) =

• { (−1)n−i χi(G) λi : i ∈ {1, . . . , n} }.
• 3. Poset (Gm

n , χ):

G χ F if χi(G) ≥ χi(F) for every i ∈ {1, . . . , n}.

• 1. Theorem (A. Kelmans 1996)

Let G, G ′ ∈ Gm

n and G ′ = ♦xy(G). Then G µ G ′.

• 1. Theorem (A. Kelmans 1996)

Let G, G ′ ∈ Gm

n and G ′ = ♦xy(G). Then G µ G ′.

• 2. Theorem

(P. Csikv´ ari, 2011) Let G, G ′ ∈ Gm

n and G ′ = ♦xy(G). Then

G M G ′, G χ G ′, and G I G ′.

• 1. Theorem (A. Kelmans 1996)

Let G, G ′ ∈ Gm

n and G ′ = ♦xy(G). Then G µ G ′.

• 2. Theorem

(P. Csikv´ ari, 2011) Let G, G ′ ∈ Gm

n and G ′ = ♦xy(G). Then

G M G ′, G χ G ′, and G I G ′.

• 3. Theorem.

Let G ∈ Cm

n . Then for every

∈ {r, a, τ, L, c, A, h, M, µ, χ, I} there exists a threshold graph F ∈ Cm

n such that G F.

If, in addition, m ≤ 2n − 4, then G F m

n .

Operations on weighted graphs

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