OPERATIONS ON GRAPHS

INCREASING SOME GRAPH PARAMETERS

Alexander Kelmans

University of Puerto RicoRutgers University

May 25, 2014

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

Let Q be an operation on a graph such that

G ∈ Gmn ⇒ Q(G ) ∈ Gm

n .

2. Let (Gmn ,�) be a quasi-poset. An operation Q is called

�-increasing (�-decreasing) if

Q(G ) � G (resp., Q(G ) � G ) for every G ∈ Gmn .

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

Let Q be an operation on a graph such that

G ∈ Gmn ⇒ Q(G ) ∈ Gm

n .

2. Let (Gmn ,�) be a quasi-poset. An operation Q is called

�-increasing (�-decreasing) if

Q(G ) � G (resp., Q(G ) � G ) for every G ∈ Gmn .

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. Chvatal, 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. 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. Chvatal, 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. 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. Chvatal, 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) bethe 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 newedges 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 Cmn be the set of connected graphs with n vertices and m

edges.

Claim. For every pair (n,m) of integers such that Cmn 6= ∅

there exists a unique triple (k , r , s) of integers

such that k ≥ 0, 0 ≤ r < s, and F (k , r , s) ∈ Cmn .

We call F (k , r , s) = F mn the extreme graph in Cm

n .

1. Let k , r , s be integers, k ≥ 0, and 0 ≤ r < s. Let F (k , r , s) bethe 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 newedges 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 Cmn be the set of connected graphs with n vertices and m

edges.

Claim. For every pair (n,m) of integers such that Cmn 6= ∅

there exists a unique triple (k , r , s) of integers

such that k ≥ 0, 0 ≤ r < s, and F (k , r , s) ∈ Cmn .

We call F (k , r , s) = F mn the extreme graph in Cm

n .

1. Theorem (A.K. 1970). Let n and m be natural numbers andn ≥ 3.

(a1) If n− 1 ≤ m ≤ 2n− 4, then F mn is the only threshold graph

with n vertices and m edges, i.e. Fmn = {F m

n }.

(a2) If m = 2n − 3, then F mn 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 n and m be natural numbers andn ≥ 3.

(a1) If n− 1 ≤ m ≤ 2n− 4, then F mn is the only threshold graph

with n vertices and m edges, i.e. Fmn = {F m

n }.

(a2) If m = 2n − 3, then F mn 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 S r are a matching and a star with r edges. Then

Kn−E (rP1) � Kn−E (S2+(r−2)P1) � Kn−E (F ) � Kn−E (S r ),

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 ∈ Cmn , 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 .

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

F a graph with r edges and at most n vertices, and

rP1 and S r are a matching and a star with r edges. Then

Kn−E (rP1) � Kn−E (S2+(r−2)P1) � Kn−E (F ) � Kn−E (S r ),

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

G 6∈ F mn . Suppose that the ♦-operation is �-decreasing.

(a1) If m = n − 1 ≥ 3 and G 6= Pn, then Pn � G � F n−1n .

(a2) If m = n ≥ 3 and G 6= Cn, then Cn � G � F nn .

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

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

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. 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. 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 ∈ Gmn }.

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 ∈ Cmn }.

3. Problem Find a graph Amn ∈ Gm

n with the maximum number

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

as(F mn ) = max { as(G ) : G ∈ Gm

n }.

4. Problem Find a graph Bmn ∈ Gm

n with the maximum number

of spanning trees, i.e. such that

t(Bmn ) = max { t(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 ∈ Gmn }.

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 ∈ Cmn }.

3. Problem Find a graph Amn ∈ Gm

n with the maximum number

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

as(F mn ) = max { as(G ) : G ∈ Gm

n }.

4. Problem Find a graph Bmn ∈ Gm

n with the maximum number

of spanning trees, i.e. such that

t(Bmn ) = max { t(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 ∈ Gmn }.

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 ∈ Cmn }.

3. Problem Find a graph Amn ∈ Gm

n with the maximum number

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

as(F mn ) = max { as(G ) : G ∈ Gm

n }.

4. Problem Find a graph Bmn ∈ Gm

n with the maximum number

of spanning trees, i.e. such that

t(Bmn ) = max { t(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 ∈ Gmn }.

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 ∈ Cmn }.

3. Problem Find a graph Amn ∈ Gm

n with the maximum number

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

as(F mn ) = max { as(G ) : G ∈ Gm

n }.

4. Problem Find a graph Bmn ∈ Gm

n with the maximum number

of spanning trees, i.e. such that

t(Bmn ) = max { t(G ) : G ∈ Gm

n }.

1. Poset (Gmn , �r ):

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

2. Poset (Gmn , �a):

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

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then G � G ′.

1. Poset (Gmn , �r ):

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

2. Poset (Gmn , �a):

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

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then G � G ′.

1. Poset (Gmn , �r ):

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

2. Poset (Gmn , �a):

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

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then G � G ′.

1. Poset (Gmn , �r ):

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

2. Poset (Gmn , �a):

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

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then G � G ′.

1. Poset (Gmn , �r ):

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

2. Poset (Gmn , �a):

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

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then G � G ′.

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

matrix of G .

2. Poset (Gmn , �L):

G �L F if L(λ,G ) ≥ L(λ,F ) for λ ≥ n.

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ).

ThenG �L G ′, and so G �τ G ′.

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

matrix of G .

2. Poset (Gmn , �L):

G �L F if L(λ,G ) ≥ L(λ,F ) for λ ≥ n.

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ).

ThenG �L G ′, and so G �τ G ′.

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

matrix of G .

2. Poset (Gmn , �L):

G �L F if L(λ,G ) ≥ L(λ,F ) for λ ≥ n.

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ).

ThenG �L G ′, and so G �τ G ′.

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

matrix of G .

2. Poset (Gmn , �L):

G �L F if L(λ,G ) ≥ L(λ,F ) for λ ≥ n.

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ).

ThenG �L G ′, and so G �τ G ′.

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

matrix of G .

2. Poset (Gmn , �L):

G �L F if L(λ,G ) ≥ L(λ,F ) for λ ≥ n.

3. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ).

ThenG �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 }.

2. Poset (Gmn , �c):

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

3. Theorem (A.K. 1995) Let G , G ′ ∈ Gmn and G ′ = Hxy (G ).

Then G �L,c 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 }.

2. Poset (Gmn , �c):

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

3. Theorem (A.K. 1995) Let G , G ′ ∈ Gmn and G ′ = Hxy (G ).

Then G �L,c 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 }.

2. Poset (Gmn , �c):

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

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

2. Theorem (A.K. 1995)

Let G , G ′ ∈ Gmn and G ′ = Hxy (G ).

Suppose that σ is a symmetric concave function. Then

cs(G ) ≥ cs(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.

2. Theorem (A.K. 1995)

Let G , G ′ ∈ Gmn 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 ).

2. Poset (Gmn , �A):

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

3. Theorem (A.K. 1992)

Let G , G ′ ∈ Gmn and G ′ = Hxy (G ). Then G �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 (Gmn , �A):

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

3. Theorem (A.K. 1992)

Let G , G ′ ∈ Gmn and G ′ = Hxy (G ). Then G �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 (Gmn , �A):

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

3. Theorem (A.K. 1992)

Let G , G ′ ∈ Gmn and G ′ = Hxy (G ). Then G �A G ′.

1. Poset (Gmn , �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 ′ ∈ Gmn and G ′ = Hxy (G ). Then

G �h G ′.

1. Poset (Gmn , �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 ′ ∈ Gmn 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).

ThenGa �c Gb

and

v(A) < v(B) ⇒ Ga ��c Gb.

1. LetΦ(λ,G ) = λm−n L(λ,G )

and λ(G ) the maximum Laplacian eigenvalue of G .

2. Poset (Gmn , �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

on the above figure. Then Ga �φ Gb.

1. LetΦ(λ,G ) = λm−n L(λ,G )

and λ(G ) the maximum Laplacian eigenvalue of G .

2. Poset (Gmn , �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

on the above figure. Then Ga �φ Gb.

1. LetΦ(λ,G ) = λm−n L(λ,G )

and λ(G ) the maximum Laplacian eigenvalue of G .

2. Poset (Gmn , �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

on the above figure. Then Ga �φ Gb.

1. LetΦ(λ,G ) = λm−n L(λ,G )

and λ(G ) the maximum Laplacian eigenvalue of G .

2. Poset (Gmn , �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

on 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).

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

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

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

of 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.

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

of S .

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

of S .

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

of 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

of 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 ∈ Cmn , Pn an n-vertex path,

Cn an n-vertex cycle, and G 6= F mn .

(a1) If m = n − 1 ≥ 3 and G 6= Pn, then Pn �L G �L F n−1n ,

cs(Pn) > cs(G ) > cs(F n−1n ) for every s ∈ {2, . . . , n − 2}, and

cn−1(G ) = cn−1(F n−1n ) = n.

(a2) If m = n ≥ 3 and G 6= Cn, then Cn �L G �L F nn ,

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

cn−1(G ) ≥ cn−1(F nn ).

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

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

cn−1(G ) ≥ cn−1(F n+1n ).

Theorem (A.K. 1995)

Let G ∈ Cmn and G 6= F m

n .

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

G �L F mn , cs(G ) > cs(F m

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

cn−1(G ) = cn−1(F mn ).

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

such that G 6�c F mn .

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

M(x ,G ) =∑{ (−1)r µr (G ) xn−2r : r ∈ {0, . . . , bn/2c} },

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 (Gmn , �M):

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

4. Poset (Gmn , �µ):

G �µ F if µr (G ) ≥ µr (F ) for every r ∈ {0, . . . , bn/2c}.

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

M(x ,G ) =∑{ (−1)r µr (G ) xn−2r : r ∈ {0, . . . , bn/2c} },

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 (Gmn , �M):

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

4. Poset (Gmn , �µ):

G �µ F if µr (G ) ≥ µr (F ) for every r ∈ {0, . . . , bn/2c}.

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

M(x ,G ) =∑{ (−1)r µr (G ) xn−2r : r ∈ {0, . . . , bn/2c} },

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 (Gmn , �M):

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

4. Poset (Gmn , �µ):

G �µ F if µr (G ) ≥ µr (F ) for every r ∈ {0, . . . , bn/2c}.

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

M(x ,G ) =∑{ (−1)r µr (G ) xn−2r : r ∈ {0, . . . , bn/2c} },

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 (Gmn , �M):

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

4. Poset (Gmn , �µ):

G �µ F if µr (G ) ≥ µr (F ) for every r ∈ {0, . . . , bn/2c}.

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

I(x ,G ) =∑{ (−1)s is(G ) x s : 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 (Gmn , �I ):

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

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

I(x ,G ) =∑{ (−1)s is(G ) x s : 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 (Gmn , �I ):

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

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

I(x ,G ) =∑{ (−1)s is(G ) x s : 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 (Gmn , �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.

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 (Gmn , �χ):

G �χ F if χi (G ) ≥ χi (F ) for every 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 (Gmn , �χ):

G �χ F if χi (G ) ≥ χi (F ) for every 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 (Gmn , �χ):

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

1. Theorem (A. Kelmans 1996)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then G �µ G ′.

2. Theorem (P. Csikvari, 2011)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then

G �M G ′, G �χ G ′, and G �I G ′.

3. Theorem. Let G ∈ Cmn . Then for every

� ∈ {�r , �a, �τ , �L, �c , �A, �h, �M , �µ, �χ, �I}

there exists a threshold graph F ∈ Cmn such that G � F .

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

1. Theorem (A. Kelmans 1996)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then G �µ G ′.

2. Theorem (P. Csikvari, 2011)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then

G �M G ′, G �χ G ′, and G �I G ′.

3. Theorem. Let G ∈ Cmn . Then for every

� ∈ {�r , �a, �τ , �L, �c , �A, �h, �M , �µ, �χ, �I}

there exists a threshold graph F ∈ Cmn such that G � F .

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

1. Theorem (A. Kelmans 1996)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then G �µ G ′.

2. Theorem (P. Csikvari, 2011)

Let G , G ′ ∈ Gmn and G ′ = ♦xy (G ). Then

G �M G ′, G �χ G ′, and G �I G ′.

3. Theorem. Let G ∈ Cmn . Then for every

� ∈ {�r , �a, �τ , �L, �c , �A, �h, �M , �µ, �χ, �I}

there exists a threshold graph F ∈ Cmn such that G � F .

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

Operations on weighted graphs

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