Let \(G\) and \(F\) be graphs. If every edge of \(G\) belongs to a subgraph of \(G\) isomorphic to \(F\), and there exists a bijection \(\lambda: V(G) \bigcup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that the set \(\{\sum_{v\in V(F’)}\lambda(v)+\sum_{e\in E(f’)}\lambda(e):F’\cong F,F’\subseteq G\}\) forms an arithmetic progression starting from \(a\) and having common difference \(d\), then we say that \(G\) is \((a,d)\)-\(F\)-antimagic. If, in addition, \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then \(G\) is \emph{super} \((a,d)\)-\(F\)-antimagic. In this paper, we prove that the grid (i.e., the Cartesian product of two nontrivial paths) is super \((a,1)\)-\(C_4\)-antimagic.

Restricted edge connectivity is a more refined network reliability index than edge connectivity. It is known that communication networks with larger restricted edge connectivity are more locally reliable.
This work presents a distance condition for graphs to be maximally restricted edge connected, which generalizes Plesník’s corresponding result.

Murty characterized the connected binary matroids with all circuits having the same size. Here we characterize the connected
bicircular matroids with all circuits having the same size.

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.