Post on 15-Mar-2020
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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