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.

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.

The Laplacian-energy-like graph invariant of a graph \(G\), denoted by \(LEL(G)\), is defined as \(LEL(G) = \sum\limits_{i=1}^{n} \sqrt{\mu_i}\), where \(\mu_i\) are the Laplacian eigenvalues of graph \(G\). In this paper, we study the maximum \(LEL\) among graphs with a given number of vertices and matching number. Some results on \(LEL(G)\) and \(LEL(\overline{G})\) are obtained.

In this paper, we consider mixed arrangements, which are composed of
hyperplanes (or subspaces) and spheres. We investigate the posets of
their intersection sets and calculate the Möbius functions of the
mixed arrangements through the hyperplane (or subspace) arrangements’
Möbius functions. Furthermore, by employing the method of deletion
and restriction, we derive recursive formulas for the triples of
these mixed arrangements.

For every integer \(c\), let \(n = R_d(c)\) be the least integer such
that for every coloring \(\Delta: \{1, 2, \ldots, 2n\} \to \{0, 1\}\),
there exists a solution \((x_1, x_2, x_3)\) to
\[x_1 + x_2 + x_3 = c\]
such that \(x_i \neq x_j\) when \(i \neq j\),
and
\(\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\).

In this paper, it is shown that for every integer \(c\),
\[R_d(c) =
\begin{cases}
4c + 8 & \text{if } c \geq 1,\\
8 & \text{if } -3 \leq c < -6,\\ 9 & \text{if} c=0,-2,-7,-8\\ 10 & \text{if } c =-1,-9 \\ |c| -\left\lfloor \frac{|c|-4}{5} \right\rceil & \text{if } c \leq -10. \end{cases}\]

A graph \(G\) with an even number of vertices is said to be
almost self-complementary if it is isomorphic to one of its
almost complements \(G^c – M\), where \(M\) denotes a perfect matching
in its complement \(G^c\). In this paper, we show that the diameter
of connected almost self-complementary graphs must be \(2\), \(3\), or
\(4\). Furthermore, we construct connected almost self-complementary
graphs with \(2n\) vertices having diameter \(3\) and \(4\) for each \(n \geq 3\),
and diameter \(2\) for each \(n \geq 4\), respectively. Additionally, we
also obtain that for any almost self-complementary graph \(G_n\) with
\(2n\) vertices, \(\lceil \sqrt{n}\rceil \leq \chi(G_n) \leq n\). By
construction, we verify that the upper bound is attainable for each
positive integer \(n\), as well as the lower bound when \(\sqrt{n}\)
is an integer.

A \(k\)-container \(C(u,v)\) in a graph \(G\) is a set of \(k\) internally
vertex-disjoint paths between vertices \(u\) and \(v\). A \(k^*\)-container
\(C(u,v)\) of \(G\) is a \(k\)-container such that \(C(u,v)\) contains all
vertices of \(G\). A graph is globally \(k^*\)-connected if there exists
a \(k^*\)-container \(C(u,v)\) between any two distinct vertices \(u\) and \(v\).
A \(k\)-regular graph \(G\) is super \(k\)-spanning connected if \(G\) is
\(i^*\)-connected for \(1 \leq i \leq k\). A graph \(G\) is \(1\)-fault-tolerant
Hamiltonian if \(G – F\) is Hamiltonian for any \(F \subseteq V(G)\) and
\(|F| = 1\). In this paper, we prove that for cubic graphs, every
super \(3\)-spanning connected graph is globally \(3^*\)-connected and
every globally \(3^*\)-connected graph is \(1\)-fault-tolerant Hamiltonian.
We present examples of super \(3\)-spanning connected graphs, globally
\(3^*\)-connected graphs that are not super \(3\)-spanning connected,
\(1\)-fault-tolerant Hamiltonian graphs that are globally \(1^*\)-connected
but not globally \(3^*\)-connected, and \(1\)-fault-tolerant Hamiltonian
graphs that are neither globally \(1^*\)-connected nor globally \(3^*\)-connected.
Furthermore, we prove that there are infinitely many graphs in each
such family.

In this paper, we prove a fixed point theorem for weakly compatible mappings satisfying a general contractive condition of operator type. In short, we are going to study mappings \( A, B, S, T: X \to X \) for which there exists a right continuous function \( \psi: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \(\psi(0) = 0\) and \(\psi(s)\leq s\) for \(s > 0.\) Moreover, for each \( x, y \in X \), one has \(O(f; d(Sx, Ty)) \leq \psi(O(f; M(x,y))),\) where \( O(f; \cdot) \) and \( f \) are defined in the first section. Also in the first section, we give some examples for \( O(f; \cdot) \). The second section contains the main result. In the last section, we give some corollaries and remarks.

We consider unitary graphs attached to \(\mathbb{Z}^{d}_{n}\) using an analogue of the Euclidean distance. These graphs are shown to be integral when \(d\) is odd or the dimension \(d\) is even.

A graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we determine the graph with the
largest spectral radius among all connected cactuses with n vertices and edge independence number \(q\).

In this study, we obtained lower and upper bounds for the Euclidean norm of a complex matrix \(A\) of order \(n \times n\). In addition,
we found lower and upper bounds for the spectral norms and Euclidean norms of the Hilbert matrix its Hadamard
square root, Cauchy-Toeplitz and Cauchy-Hankel matrices in the forms \(H = \left(\frac{1}{i + j – 1}\right)_{i,j=1}^n\),\(H^{\frac{01}{2}}=(\frac{1}{(i+j-1)}^{\frac{1}{2}})_{i,j=1}^n\); \(T_n = \left[\frac{1}{(g+(i + j)h)}_{i,j=1}^n\right]\), and \(H_n = \left[\frac{1}{(g+(i + j )h}\right]_{i,j=1}^n\), respectively.

Let \(G\) be a graph, and let \(a\) and \(b\) be nonnegative integers such that \(1 \leq a \leq b\). Let \(g\) and \(f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \((g, f)\)-factor if \(g(x) \leq d_G^h(x) \leq f(x)\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy \in E(G)\}\). The isolated toughness \(I(G)\) of a graph \(G\) is defined as follows: If \(G\) is a complete graph, then \(I(G) = +\infty\); else, \(I(G) = \min\{ \frac{|S|}{i(G-S)} : S \subseteq V(G), i(G – S) \geq 2 \}\), where \(i(G – S)\) denotes the number of isolated vertices in \(G – S\). In this paper, we prove that \(G\) has a fractional \((g, f)\)-factor if \(\delta(G) \geq I(G) \geq \frac{b(b-1)}{a}+1\). This result is best possible in some sense.

In this paper we prove that there exists one type of connected cubic graph,which minimizes the number of spanning trees over all other connected cubic graphs of the same order \(7\), \(n\geq 14\).

Let \(T = PSL(n, q)\) be a projective linear simple group, where \(n \geq 2\),\(q\) a prime power and \((n,q) \neq (2,2)\) and \((2,3)\). We classify all \(3— (v, k, 1)\) designs admitting an automorphism group \(G\) with \(T \unlhd G \leq Aut(T)\) and \(v=\frac{q^n-1}{q-1}.\)

In this paper, we introduce the notion of \(f\)-derivations and investigate the properties of \(f\)-derivations of lattice implication
algebras. We provide an equivalent condition for an isotone \(f\)-derivation in a lattice implication algebra. Additionally, we
characterize the fixed set \({Fix_d}(L)\) and \(\mathrm{Kerd}\) by \(f\)-derivations. Furthermore, we introduce
normal filters and obtain some properties of normal filters in lattice implication algebras.

We give a new combinatorial interpretation of Lah and \(r\)-Lah numbers.
We establish two cross recurrence relations: the first one, which uses
an algebraic approach, is a recurrence relation of order two with
rational coefficients; the second one uses a combinatorial proof and
is a recurrence relation with integer coefficients. We also express
\(r\)-Lah numbers in terms of Lah numbers. Finally, we give identities
related to rising and falling factorial powers.

In this paper, we reveal the yin-yang structure of the affine plane of order four by characterizing the unique blocking set as the
Mébius-Kantor configuration \(8_3\).

A family of sets is called \(K\)-union distinct if all unions involving \(K\) or fewer members thereof are distinct. If a family of
sets is \(K\)-cover-free, then it is \(K\)-union distinct. In this paper, we recognize that this is only a sufficient condition and,
from this perspective, consider partially cover-free families of sets with a view to constructing union distinct families. The
role of orthogonal arrays and related combinatorial structures is explored in this context. The results are applied to find
efficient anti-collusion digital fingerprinting codes.

Let \(G\) be a \(2\)-edge-connected simple graph on \(n\) vertices, \(n \geq 3\). It is known that if \(G\) satisfies \(d(x) \geq \frac{n}{2}\) for every vertex \(x \in V(G)\), then \(G\) has a nowhere-zero \(3\)-flow, with several exceptions.In this paper, we prove that, with ten exceptions, all graphs with at most two vertices of degree less than \(\frac{n}{2}\) have nowhere-zero \(3\)-flows. More precisely, if \(G\) is a \(2\)-edge-connected graph on \(n\) vertices, \(n \geq 3\), in which at most two vertices have degree less than \(\frac{n}{2}\), then \(G\)
has a nowhere-zero \(3\)-flow if and only if \(G\) is not one of ten completely described graphs.

In this paper, we introduce the notion of right derivation of a weak BCC-algebra and investigate its related properties.
Additionally, we explore regular right derivations and d-invariants on weak BCC-ideals in weak BCC-algebras.

We investigate the Jacobsthal numbers \(\{J_n\}\) and Jacobsthal-Lucas numbers \(\{j_n\}\). Let \(\mathcal{J}_n = J_n \times j_n\) and \(\mathcal{J}_n = J_n + j_n\).In this paper, we give some determinantal and permanental representations for \(\mathcal{J}_n\) and \(\mathcal{J}_n\). Also, complex factorization formulas for the numbers are presented.

Let \(d\) be a fixed integer, \(0 \leq d \leq 2\), and let \(\mathcal{K}\) be a family of sets in the plane having simply connected union. Assume that for every countable subfamily \(\{K_n : n \geq 1\}\) of \(\mathcal{K}\), the union \(\cup\{K_n \geq 1\}\) is
starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K:K \in \mathcal{K}\}\) has these properties as well.
In the finite case ,define function \(g\) on \((0, 1, 2) \) by \(g(0) = 2\), \(g(1) = g(2) = 4\). Let \(\mathcal{K}\) be a finite family of nonempty compact sets in the plane such that \(\cup\{K \in \mathcal{K}\}\) has a connected complement. For fixed \(d \in \{0, 1, 2\}\), assume that for every \(g(d)\) members of \(\mathcal{K}\), the corresponding union is starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K \in \mathcal{K}\}\) also has these properties,also.
Most of these results are dual versions of theorems that hold for intersections of sets starshaped via staircase paths.The exceotion is the finite case above when \(d = 2\) .Surprisingly ,although the result for \(d=2\) holds for unique of sets, no analogue for intersections of sets is possible.

Let \(G\) be a simple connected graph containing a perfect matching.
\(G\) is said to be BM-extendable (bipartite matching extendable)
if every matching \(M\) which is a perfect matching of an induced
bipartite subgraph of \(G\) extends to a perfect matching of \(G\).

The BM-extendable cubic graphs are known to be \(K_{4}\) and \(K_{3,3}\).
In this paper, we characterize the 4-regular BM-extendable graphs.
We show that the only 4-regular BM-extendable graphs are \(K_{4,4}\) and
\(T_{4n}\), \(n \geq 2\), where \(T_{4n}\) is the graph on \(4n\) vertices
\(u_{i}\), \(v_{i}\), \(x_{i}\), \(y_{i}\), \(1 \leq i \leq n\), such that
\(\{u_{i}, v_{i}, x_{i}, y_{i}\}\) is a clique and
\(x_{i}u_{i+1}\), \(y_{i}v_{i+1} \in E(T_{4n})\) (mod \(n\)).

A rainbow coloring of the edges of a graph is a coloring such
that no two edges of the graph have the same color. The
anti-Ramsey number \(f(G, H)\) is the maximum number of colors
such that there is an \(H\)-anti-Ramsey edge coloring of \(G\), that is,
there exists no rainbow copy of the subgraph \(H\) of \(G\) in some
coloring of the edges of the host graph \(G\) with \(f(G, H)\) colors.

In this note, we exactly determine \(f(Q_5, Q_2)\) and \(f(Q_5, Q_3)\),
where \(Q_n\) is the \(n\)-dimensional hypercube.

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum
of weights \(\frac{2}{d(u) + d(v)}\) of all edges \(uv\) of \(G\), where
\(d(u)\) denotes the degree of a vertex \(u\) in \(G\).

In this paper, we establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.

For a graph \(G\), its Hosoya index is defined as the total number
of matchings in it, including the empty set. As one of the oldest and
well-studied molecular topological descriptors, the Hosoya index has
been extensively explored.

Notably, existing literature has primarily focused on its extremal
properties. In this note, we bridge a significant gap by establishing
sharp lower bounds for the Hosoya index in terms of other topological
indices.

We present a unified extension of alternating subsets to \(k\)-combinations
of \(\{1, 2, \ldots, n\}\) containing a prescribed number of sequences
of elements of the same parity. This is achieved by shifting attention
from parity-alternating elements to pairs of adjacent elements of the
same parity.

Enumeration formulas for both linear and circular combinations are
obtained by direct combinatorial arguments. The results are applied
to the enumeration of bit strings.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\).
Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\),
where \(d(D)\) denotes the diameter of the digraph \(D\).

In this paper, it is shown that \(\overrightarrow{d}(G(n_1, n_2, \ldots, n_p)) = d(G)\),
where \(G(n_1, n_2, \ldots, n_p)\) is a \(G\)-vertex multiplication
([2]) of a connected bipartite graph \(G\) of order \(p \geq 3\)
with diameter \(d(G) \geq 5\) and any finite sequence \(\{n_1, n_2, \ldots, n_p\}\)
with \(n_i \geq 3\).

Cyclic frames, or partially partition-type cyclic relative difference
families, are combinatorial structures that are used to produce series
of optimal families consisting of a single frequency hopping sequence
and optimal difference systems of sets for code synchronization.

In this paper, two new classes of cyclic frames from finite geometries
are obtained.

Consider the game of locating a marked vertex on a connected graph,
where the player repeatedly chooses a vertex of the graph as a probe,
and is given the distance from the probe to the marked vertex,
until she can uniquely locate the hidden vertex. The goal is to
minimize the number of probes.

The static version of this game is the well-known problem of finding
the metric dimension (or location number ) of the graph.
We study the sequential version of this game, and the corresponding
sequential location number .

We establish several formulae for sums and alternating sums of products
of generalized Fibonacci and Lucas numbers. In particular, we extend
some results of Z. Cerin and of Z. Cerin and G. M. Gianella .

An \({H}_2\) graph is a multigraph on three vertices with a double
edge between a pair of distinct vertices and single edges between
the other two pairs. In this paper, we settle the \({H}_2\) graph
decomposition problem, which was left unfinished in a paper of
Hurd and Sarvate, by decomposing a complete multigraph \(3K_{8t}\)
into \({H}_2\) graphs recursively.

This article is a contribution to the study of the automorphism groups
of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design and
suppose that \(G\) is a group of automorphisms of \(\mathcal{D}\) which is
block-transitive and point-primitive. Then \(\mathrm{Soc}(G)\),
the socle of \(G\), is not isomorphic to \(^2G_2(q)\) or to \(^2F_4(q^2)\)
for any prime power \(q\).

Let \(G\) be a finite permutation group acting primitively on sets \(\Omega_1\) and \(\Omega_2\). We describe a construction of a \(1\)-design
with the block set \(\mathcal{B}\) and the point set \(\Omega_2\), having \(G\) as an automorphism group.Applying this method, we construct a unital \(2\)-\((q^3+1, q+1, 1)\) design and a semi-symmetric design \((q^4-q^3+q^2, q^2-q, (1))\) from the unitary group \(U(3,q)\), where \(q = 3, 4, 5, 7\).From the unital and the semi-symmetric design, we build a projective plane \(PG(2,q^2)\). Further, we describe other combinatorial structures constructed from these unitary groups.