Let \(D\) be a strongly connected digraph with order at least two. Let \(M(D)\) denote the middle digraph of \(D\), and let \(\kappa(D)\) and \(\lambda(D)\) denote the connectivity and arc-connectivity of \(D\), respectively. In this paper, we study super-arc-connected and super-connected middle digraphs and the spectrum of middle digraphs.

For a connected graph \(G\) of order \(p \geq 2\), a set \(S \subseteq V(G)\) is an \(x\)-geodominating set of \(G\) if each vertex \(v \in V(G)\) lies on an \(x\)-geodesic for some element \(y \in S\). The minimum cardinality of an \(x\)-geodominating set of \(G\) is defined as the \(\alpha\)-geodomination number of \(G\), denoted by \(g_x(G)\) or simply \(g_x(G)\). An \(x\)-geodominating set of cardinality \(g_x(G)\) is called a \(g_x(G)\)-set. A connected graph of order \(p\) with vertex geodomination numbers either \(p – 1\) or \(p – 2\) for every vertex is characterized. It is shown that there is no graph of order \(p\) with vertex geodomination number \(p – 2\) for every vertex. Also, for an even number \(p\) and an odd number \(n\) with \(1 \leq n \leq p – 1\), there exists a connected graph \(G\) of order \(p\) and \(g_x(G) = n\) for every vertex \(x \in G\), and for an odd number \(p\) and an even number \(n\) with \(1 \leq n \leq p – 1\), there exists a connected graph \(G\) of order \(p\) and \(g_x(G) = n\) for every vertex \(x \in G\). It is shown that for any integer \(n > 2\), there exists a connected regular as well as a non-regular graph \(G\) with \(g_x(G) = n\) for every vertex \(x \in G\). For positive integers \(r, d\) and \(n \geq 2\) with \(r \leq d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(g_x(G) = n\) for every vertex \(x \in G\). Also, for integers \(p, d\) and \(n\) with \(3 \leq d \leq p – 1, 1 \leq n \leq p – 1\) and \(p – d – n + 1 \geq 0\), there exists a graph \(G\) of order \(p\), diameter \(d\) and \(g_x(G) = n\) for some vertex \(x \in G\).

A graph is called \emph{biclaw-free} if it has no biclaw as an induced subgraph. Lai and Yao [Discrete Math., \(307 (2007) 1217\)] conjectured that every \(2\)-connected biclaw-free graph \(G\) with \(\delta(G) \geq 4\) has a spanning eulerian subgraph \(H\) with maximum degree \(\Delta(H) \leq 4\). In this note, the conjecture is answered in the negative.

Let \(G\) be a graph of order \(n\) and size \(m\). A \(\gamma\)-labeling of \(G\) is a one-to-one function \(f: V(G) \to \{0, 1, 2, \ldots, m\}\) that induces a labeling \(f’: E(G) \to \{1, 2, \ldots, m\}\) of the edges of \(G\) defined by \(f'(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined as

\[val(f) = \sum\limits_{e \in E(G)} f'(e).\]

The \(\gamma\)-spectrum of a graph \(G\) is defined as

\[spec(G) = \{val(f): f \text{ is a \(\gamma\)-labeling of } G\}.\]

The \(\gamma\)-spectra of paths, cycles, and complete graphs are determined.

An \((a, d)\)-edge-antimagic total labeling on a \((p, q)\)-graph \(G\) is a one-to-one map \(f\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, p+q\) with the property that the edge-weights, \(w(uv) = f(u) + f(v) + f(uv)\) where \(uv \in E(G)\), form an arithmetic progression starting from \(a\) and having common difference \(d\). Such a labeling is called \emph{super} if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labeling of the disjoint union of multiple copies of the complete tripartite graph and the disjoint union of stars.

Given a configuration of pebbles on the vertices of a graph \(G\), a pebbling move consists of taking two pebbles off a vertex \(v\) and putting one of them back on a vertex adjacent to \(v\). A graph is called \({pebbleable}\) if for each vertex \(v\) there is a sequence of pebbling moves that would place at least one pebble on \(v\). The \({pebbling\;number}\) of a graph \(G\), is the smallest integer \(m\) such that \(G\) is pebbleable for every configuration of \(m\) pebbles on \(G\). A graph \(G\) is said to be class \(0\) if the pebbling number of \(G\) is equal to the number of vertices in \(G\). We prove that \(Bi-wheels\), a class of diameter three graphs, are class \(0\).

In this paper, we study the flexibility of embeddings of circular graphs \(C(2n,2)\), \(n \geq 3\) on the projective plane. The numbers of (non-equivalent) embeddings of \(C(2n, 2)\) on the projective plane are obtained, and by describing structures of these embeddings, the numbers of (non-equivalent) weak embeddings and strong embeddings of \(C(2n, 2)\) on the projective plane are also obtained.

In \([4]\), Elizalde and Pak gave a bijection \(\Theta: S_n(321) \to S_n(132)\) that commutes with the operation of taking inverses and preserves the numbers of fixed points and excedances for every \(\Gamma \in S_n(321)\). In \([1]\) it was shown that another bijection \(\Gamma: S_n(321) \to S_n(132)\) introduced by Robertson in \([7]\) has these same properties, and in \([2]\) a pictorial reformulation of \(\Gamma\) was given that made it clearer why \(\Gamma\) has these properties. Our purpose here is to give a similar pictorial reformulation of \(\Theta\), from which it follows that, although the original definitions of \(\Theta\) and \(\Gamma\) make them appear quite different, these two bijections are in fact related to each other in a very simple way, by using inversion, reversal, and complementation.

Gyarfas conjectured that for a given forest \(F\), there exists an integer function \(f(F,w(G))\) such that \(\chi(G) \leq f(F,w(G))\) for any \(F\)-free graph \(G\), where \(\chi(G)\) and \(w(G)\) are respectively, the chromatic number and the clique number of G. Let G be a \(C_5\)-free graph and \(k\) be a positive integer. We show that if \(G\) is \((kP_1, + P_2)\)-free for \(k \geq 2\), then \(\chi(G) \leq 2w^{k-1} \sqrt{w}\); if \(G\) is \((kP_1, + P_3)\)-free for \(k \geq 1\), then \(\chi(G) \leq w^k \sqrt{w}\). A graph \(G\) is \(k\)-divisible if for each induced subgraph \(H\) of \(G\) with at least one edge, there is a partition of the vertex set of \(H\) into \(k\) sets \({V_1,… , V_k}\) such that no \(V_i\); contains a clique of size \(w(G)\). We show that a \((2P_1+P_2)\)-free and \(C_5\)-free graph is \(2\)-divisible.

The concept of the sum graph and integral sum graph were introduced by F. Harary. Let \(\mathbb{N}\) denote the set of all positive integers. The sum graph \(G^+(S)\) of a finite subset \(S \subset {N}\) is the graph \((S, E)\) with \(uv \in E\) if and only if \(u+v \in S\). A simple graph \(G\) is said to be a sum graph if it is isomorphic to a sum graph of some \(S \subset {N}\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which when added to \(G\) result in a sum graph. Let \(\mathbb{Z}\) denote the set of all integers. The integral sum graph \(G^+(S)\) of a finite subset \(S \subset {Z}\) is the graph \((S, E)\) with \(uv \in E\) if and only if \(u+v \in S\). A simple graph \(G\) is said to be an integral sum graph if it is isomorphic to an integral sum graph of some \(S \subset {Z}\). The integral sum number \(\zeta(G)\) of \(G\) is the smallest number of isolated vertices which when added to \(G\) result in an integral sum graph. In this paper, we investigate and determine the sum number and the integral sum number of the graph \(K_n \setminus E(C_{n-1})\). The results are presented as follows:\(\zeta(K_n \setminus (C_{n-1})) = \begin{cases}
0, & n = 4,5,6,7 \\
2n-7, & n \geq 8
\end{cases}\)
and
\(\sigma(K_n \setminus E(C_{n-1})) = \begin{cases}
1, & n = 4 \\
2, & n = 5\\
5, & n = 5\\
7, & n = 7\\
2n-7, & n \geq 8
\end{cases}\)