An \(L(2,1)\)-labeling of a graph \(G\) is an assignment of nonnegative
integers to the vertices of \(G\) such that adjacent vertices get numbers
at least two apart, and vertices at distance two get distinct numbers.
The \(L(2,1)\)-labeling number of \(G\), \(\lambda(G)\), is the minimum range of
labels over all such labelings. In this paper, we determine the \(\lambda\)-
numbers of flower snark and its related graphs for all \(n \geq 3\).

In this paper, some limit relations between multivariable
Hermite polynomials \((MHP)\) and some other multivariable polyno-
mials are given, a class of multivariable polynomials is defined via
generating function, which include \((MHP)\) and multivariable Gegen-
bauer polynomials \((MGP)\) and with the help of this generating func-
tion various recurrence relations are obtained to this class. Integral
representations of \(MHP\) and \(MGP\) are also given. Furthermore, gene-
ral families of multilinear and multilateral generating functions are
obtained and their applications are presented.

We give some properties of skew spectrum of a graph, especially,
we answer negatively a problem concerning the skew characteristic
polynomial and matching polynomial in [M. Cavers et al., Skew-
adjacency matrices of graphs, Linear Algebra Appl. \(436 (2012) 4512-
4529]\).

This paper is devoted to studying the form of the solutions and
the periodicity of the following rational system of difference
equations:

\begin{align*}
x_{n+1} &= \frac{x_{n-5}}{1-x_n-_5y_{n-2}}, &
y_{n+1}= \frac{ y_{n-5}}{\pm1 \pm y_{n-5} + _5x_{n-2}},
\end{align*}

with initial conditions are real numbers.

The Moore bound states that a digraph with maximum out-degree \(d\)
and radius \(k\) has at most \(1 + d + \cdots + d^k\) vertices.
Regular digraphs attaining this bound and whose diameter is at most
\(k + 1\) are called radially Moore digraphs. Körner [4] proved
that these extremal digraphs exist for any value of \(d \geq 1\) and \(k \geq 1\).

In this paper, we introduce a digraph operator based on the line
digraph, which allows us to construct new radially Moore digraphs
and recover the known ones. Furthermore, we show that for \(k = 2\),
a radially Moore digraph with as many central vertices as the degree
\(d\) does exist.

The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\)
is the set consisting of \(e\) and of all edges having a common
end-vertex with \(e\) . Let \(f\) be a function on \(E(G)\) , the edge
set of \(G\) , into the set \(\{-1, 0, 1\}\). If \(\sum_{x \in N_G[e]} f(x) \geq 1\)
for each \(e \in E(G)\), then \(f\) is called a minus edge
dominating function of \(G\).

The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over
all minus edge dominating functions \(f\) of \(G\), is called the
\emph{minus edge domination number} of \(G\) and is denoted by
\(\gamma’_m(G)\).

It has been conjectured that \(\gamma’_m(G) \geq n – m\) for every
graph \(G\) of order \(n\) and size \(m\). In this paper, we prove
that this conjecture is true and then classify all graphs \(G\)
with \(\gamma’_m(G) = n – m\).

We seek a decomposition of a complete equipartite graph minus
a one-factor into parallel classes each consisting of cycles of length
\(k\). In this paper, we address the problem of resolvably decomposing
complete multipartite graphs with \(r\) parts each of size \(\alpha\) with a one-
factor removed into \(k\)-cycles. We find the necessary conditions, and
give solutions for even cycle lengths.

An adjacent vertex distinguishing edge coloring, or an avd-coloring,
of a simple graph \(G\) is a proper edge coloring of \(G\) such that
no two adjacent vertices are incident with the same set of colors.

H. Hatami showed that every simple graph \(G\) with no isolated
edges and maximum degree \(\Delta\) has an avd-coloring with at
most \(\Delta + 300\) colors, provided that \(\Delta > 10^{20}\).

We improve this bound as follows: if \(\Delta > 10^{15}\), then the
avd-chromatic number of \(G\) is at most \(\Delta + 180\), where
\(\Delta\) is the maximum degree of \(G\).

The Padmakar-Ivan (\(PI\)) index of a graph \(G = (V, E)\) is defined
as \(PI(G) = \sum_{e \in uv} (n_{eu}(e|G) + n_{ev}(e|G))\)
where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\)
than to \(v\) and \(n_{ev}(e|G)\) is the number of edges of \(G\) lying
closer to \(v\) than to \(u\).

In this paper, we derive a recursive formula for computing the
\(PI\) index of a double hexagonal chain using the orthogonal cut,
and characterize the double hexagonal chains with extremal
\(PI\) indices.

In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is
performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We define the
pegging number as the smallest number of pegs needed to reach all the vertices in a graph no matter what the distribution. Similarly, the optimal-pegging number of a graph is defined as the smallest distribution of pegs for which all the vertices in the graph can be reached.We obtain tight bounds on the pegging numbers and optimal-pegging numbers of complete binary trees and compute the optimal-pegging numbers of complete infinitary trees. As a result of these computaions, we deduce that there is a tree whose optimal-pegging number is strictly increased by removing a leaf. We also compute the optimal-pegging number of caterpillar graphs and the tightest upper boundon the optimal-pegging numbers of lobster graphs.