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.