Let $G=(V,E)$ be a graph. If $G$ is a König graph or if $G$ is a graph without 3-cycles and 5-cycles, we prove that the following conditions are equivalent: $\Delta_{G}$ is pure shellable, ...$R/I_{\Delta}$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of König type $e_{1},\dots,e_{g}$ without 4-cycles with two $e_i$'s. Furthermore, we study vertex decomposable and shellable (non-pure) properties in graphs without 3-cycles and 5-cycles. Finally, we give some properties and relations between critical, extendable and shedding vertices.
This article studies the multi-layer and multi-pass welding of the circumferential weld in the thick pipe welding of an aviation enterprise, based on the static characteristics of the welding arc, ...through data calculations and experiments to verify the relationship between arc voltage and arc length, proposes to use the arc voltage controller to accurately adjust the arc voltage in real time, improve the welding stability, use the welding torch wiggler to expand the welding fusion width, meet the adaptive adjustment of the wide weld, and build the main control system based on programmable logic controller (PLC) control. Integrating all subsystems and put them into the actual production of the enterprise, significantly improves the production efficiency of parts welding, reduces welding defects, and improves the quality of aviation parts.
The PI index of a graph G is defined by PI(G)=∑e=(u,v)∈Emu(e|G)+mv(e|G) where mu(e|G) be the number of edges in G lying closer to vertex u than to vertex v and mv(e|G) be the number of edges in G ...lying closer to vertex v than to vertex u. In this paper, we give the upper and lower bounds on the PI index of connected unicyclic and bicyclic graphs with given girth and completely characterize the corresponding extremal graphs. From our results, it is easy to get the bounds and extremal graphs of the unicyclic and bicyclic graphs.
In this paper, we propose a new multi-edge metric-constrained quasi-cyclic progressive edge-growth algorithm (MM-QC-PEGA), which is suitable for constructing both single- and multi-weighted (binary) ...QC low-density parity-check (LDPC) codes with arbitrary lengths, rates, circulant sizes, and variable node (VN)-degree distributions. The MM-QC-PEGA is able to detect all cyclic-edge-set-minimum-virtual cycles (CMVCs), as it accurately computes the metric value of each CMVC with an a posteriori test. In addition, we propose a new greatest-common-divisor (GCD)-approximation for time efficiently approximating the metric value of a CMVC without a posteriori tests, and propose a new GCD-approximated MM-QC-PEGA (G-MM-QC-PEGA) by using the GCD-approximation to replace the a posteriori test involved in the MM-QC-PEGA. As a result, the G-MM-QC-PEGA is faster than the MM-QC-PEGA, and the CMVCs undetectable to the G-MM-QC-PEGA under multi- and single-weighted QC-LDPC code graphs have minimum lengths of eight and ten, respectively. Moreover, we propose a new masking technique that is efficient in masking the parity-check matrix consisting of an array of arbitrary circulants of the same size. Compared with the several existing works, our proposed algorithms could perform better or comparably in terms of avoiding generating small cycles and error performances. Our proposed algorithms somewhat perfect the works of QC-LDPC code construction.
The isometric Ramsey number IR(H→) of a family H→ of digraphs is the smallest number of vertices in a graph G such that any orientation of the edges of G contains every member of H→ in the ...distance‐preserving way. We observe that the isometric Ramsey number of a finite family of finite acyclic digraphs is always finite, and present some bounds in special cases. For example, we show that the isometric Ramsey number of the family of all oriented trees with n vertices is at most n2n+o(n).
The rings considered in this article are commutative with identity which are not integral domains. Let
R
be a ring. Recall that an element
x
of
R
is an exact zero-divisor if there exists a non-zero ...element
y
of
R
such that
A
n
n
(
x
)
=
R
y
and
A
n
n
(
y
)
=
R
x
. As in Lalchandani (International J. Science Engineering and Management (IJSEM) 1(6): 14-17, 2016), for a ring
R
, we denote the set of all exact zero-divisors of
R
by
EZ
(
R
) and
E
Z
(
R
)
\
{
0
}
by
E
Z
(
R
)
∗
. Let
R
be a ring. In the above mentioned article, Lalchandani introduced and studied the properties of a graph denoted by
E
Γ
(
R
)
, which is an undirected graph whose vertex set is
E
Z
(
R
)
∗
and distinct vertices
x
and
y
are adjacent in
E
Γ
(
R
)
if and only if
A
n
n
(
x
)
=
R
y
and
A
n
n
(
y
)
=
R
x
. Let
R
be a reduced ring such that
E
Z
(
R
)
∗
≠
∅
. The aim of this article is to study the interplay between the graph-theoretic properties of
E
Γ
(
R
)
and the ring-theoretic properties of
R
.
For integer k≥2 and prime power q, Lazebnik and Ustimenko (1995) proposed an algebraic bipartite graph D(k,q) which is q-regular, edge-transitive and of large girth. Füredi et al. (1995) conjectured ...that D(k,q) has girth k+5 for all odd k and all q≥4 and, shown that this conjecture is true for the case that (k+5)/2 divides q−1. Cheng et al. (2014) shown that this conjecture is true for the case that (k+5)/2 is an arbitrary power of the characteristic of Fq. In this paper, we propose a generalization for the binomial coefficients and show that this conjecture is true when (k+5)/2 is the product of an arbitrary factor of q−1 and an arbitrary power of the characteristic of Fq.
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.
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