We study hypergraphs which represent finite quantum event structures. We contribute to results of graph theory, regarding bounds on the number of edges, given the number of vertices. We develop a ...missing one for 3-graphs of girth 4. As an application of the graph-theoretical approach to quantum structures, we show that the smallest orthoalgebra with an empty state space has 10 atoms. Optimized constructions of an orthomodular poset and an orthomodular lattice with no group-valued measures are given. We present also a handcrafted construction of an orthoalgebra with no group-valued measure; it is larger, but its properties can be verified without a computer.
An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. Let χi(G) be the injective chromatic number of a graph G. In ...this paper, we investigate the injective coloring of planar graphs with girth 6. We improve some results of Borodin and Ivanova (2011) 1, Doyon et al. (2010) 4 and Li and Xu (2012) 6 by showing that if G is a planar graph with girth at least 6, then (1) χi(G)≤Δ+3; (2) χi(G)≤Δ+2 if Δ≥9; (3) χi(G)≤Δ+1 if Δ≥17.
Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T
Γ
(R), is the undirected graph with vertex set R and two distinct vertices x and y are ...adjacent if and only if x + y ∈ Z(R).
denotes the complement of T
Γ
(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi
2
. In this article, we characterize all commutative rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure
whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for
to be connected whenever
is connected and prove that
. It is also proved that if diam(T
Γ
(R)) = 2, then T
Γ
(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al.
1
.
G
has a list
k
-
L
(2, 1)-labeling if for any
k
-list assignment
L
, there exists a coloring
c
:
V
(
G
)
→
⋃
v
∈
V
L
(
v
)
of
G
such that
c
(
v
)
∈
L
(
v
)
for
∀
v
∈
V
(
G
)
and for
∀
u
,
v
∈
V
(
G
)
...,
|
c
(
u
)
-
c
(
v
)
|
≥
2
if
d
(
u
,
v
)
=
1
,
|
c
(
u
)
-
c
(
v
)
|
≥
1
if
d
(
u
,
v
)
=
2
.
λ
2
,
1
l
(
G
)
=
min
{
k
|
G
has a list
k
-
L
(2, 1)-labeling
}
is called the list
L
(2, 1)
-labeling number
of
G
. In this paper, we prove that for planar graph with maximum degree
Δ
≥
5
, girth
g
≥
13
and without adjacent 13-cycles,
λ
2
,
1
l
(
G
)
≤
Δ
+
3
holds. Moreover, the upper bound
Δ
+
3
is tight.
Multi-type quasi-cyclic (QC) low-density parity-check (LDPC) codes can be considered as multiple-edge protograph QC-LDPC codes having some advantages in the minimum Hamming distance bound over ...single-edge protograph codes or type-I QC-LDPC codes when the base matrices have the same size. In this paper, we investigate a class of multi-type QC-LDPC codes whose parity-check matrices contain just one blockrow of circulants and we obtain the generator matrix of such codes in general form. Using the permutation arrays and defining injection arrays, we present a new approach to construct a class of high-rate type-I QC-LDPC codes with girth 6 from the constructed 4-cycle free multi-type QC-LDPC codes. In continue, for 2 ≤ w ≤ 6, some type-w QC-LDPC codes with girth 6 are constructed explicitly such that the constructed codes are flexible in terms of rate and length. To the best of our knowledge, for w = 5,6, this is the first paper which deals with the explicit construction of type-w QC-LDPC codes with girth 6 and high rates. Moreover, for w = 3,4, the constructed type-w QC-LDPC codes have better (6,8)-cycle multiplicities than the codes with minimum achievable length recently constructed by cyclic difference families (CDFs). Simulation results show that the binary and non-binary constructed codes outperform the constituent underlying QC-LDPC codes.
In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial ...applications of graph theory to the solution of molecular problems. Its main goal is to use numerical invariants to reduce the topological structure of a molecule to a single number that characterizes its properties. Topological indices are numerical invariants associated with the chemical constitution, for the purpose of the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. They have found important application in predicting the behavior of chemical substances. The Graovac–Ghorbani (ABCGG) index is a topological descriptor that has improved predictive potential compared to analogous descriptors. It is used to model both the boiling point and melting point of molecules and is applied in the pharmaceutical industry. In the recent years, the number of publications on its mathematical properties has increased. The aim of this work is to partially solve an open problem, namely to find the structure of unicyclic graphs that minimize the ABCGG index. We characterize unicyclic graphs with even girth that minimize the ABCGG index, while we also present partial results for odd girths. As an auxiliary result, we compare the ABCGG indices of paths and cycles with an odd number of vertices.
Mader conjectured in 2010 that for any tree
T
of order
m
, every
k
-connected graph
G
with minimum degree at least
⌊
3
k
2
⌋
+
m
-
1
contains a subtree
T
′
≅
T
such that
G
-
V
(
T
′
)
is
k
...-connected. This conjecture has been proved for
k
=
1
; however, it remains open for general
k
≥
2
; for
k
=
2
, partially affirmative answers have been shown, all of which restrict the class of trees to special subclasses such as trees with at most 5 internal vertices, trees of order at most 8, trees with diameter at most 4, caterpillars, and spiders. We first extend the previously known subclass of trees for which Mader’s conjecture for
k
=
2
holds; namely, we show that Mader’s conjecture for
k
=
2
is true for the class of bifurcate quasi-unimodal caterpillars which includes every caterpillar and every tree of order
m
with diameter at least
m
-
4
. Instead of restricting the class of trees, we next consider 2-connected graphs with girth conditions. We then show that Mader’s conjecture is true for every 2-connected graph
G
with
g
(
G
)
≥
δ
(
G
)
-
8
, where
g
(
G
) and
δ
(
G
)
denote the girth of
G
and the minimum degree of a vertex in
G
, respectively. Besides, we show that for every 2-connected graph
G
with
g
(
G
)
≥
δ
(
G
)
-
7
, the lower bound of
m
+
2
on
δ
(
G
)
in Mader’s conjecture can be improved to
m
+
1
if
m
≥
10
. Moreover, the lower bound of
δ
(
G
)
-
8
(respectively,
δ
(
G
)
-
7
) on
g
(
G
) in these results can be improved to
δ
(
G
)
-
9
(respectively,
δ
(
G
)
-
8
with
m
≥
11
) if no six (respectively, four) cycles of length
g
(
G
) have a common path of length
g
(
G
)
2
-
1
in
G
. We also show that Mader’s conjecture holds for every 2-connected graph
G
with
g
∘
(
G
)
≥
δ
(
G
)
-
8
, where
g
∘
(
G
)
is the overlapping girth of
G
. Mader’s conjecture is interesting not only from a theoretical point of view but also from a practical point of view, since it may be applied to fault-tolerant problems in communication networks. Our proofs lead to
O
(
|
V
(
G
)
|
4
)
time algorithms for finding a desired subtree in a given 2-connected graph
G
satisfying the assumptions.
•Large set of J-solution for pipes and cylinders with mismatched girth welds.•Large set of h1-factors for girth welds with weld centerline surface cracks.•Improved reference stress methodology for ...cylinders with mismatched girth welds.•Introduction of an advanced FAD procedure based on fully-plastic solutions for the J-integral.•Use of more accurate FAD procedures yield much larger tolerable crack size estimates.
This work addresses an evaluation procedure for the J-integral in pipeline girth welds with circumferential surface cracks subjected to bending load for a wide range of crack geometries and weld mismatch levels based on the GE-EPRI and the reference stress framework. The study also addresses evaluation of critical flaw sizes for a subsea flowline clad pipe having an undermatched girth weld made of UNS N06625 Alloy 625. The 3-D numerical analyses provide a large set of J-solutions in cracked pipes with mismatched girth welds with implications of the potential applicability of ECA procedures in welded cracked structural components.
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