The bondage number \(b(D)\) of a digraph \(D\) is the cardinality of a smallest set of arcs whose removal from \(D\) results in a digraph with domination number greater than the domination number of \(D\). In this paper, we present some upper bounds on bondage number for oriented graphs including tournaments, and symmetric planar digraphs.

Let \(G\) be a connected graph. For a vertex \(v \in V(G)\) and an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the representation of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(r(v|\Pi) = (d(v, S_1), d(v, S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is said to be resolving if the \(k\)-vectors \(r(v|\Pi), v \in V(G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is called the partition dimension of \(G\), denoted by \(pd(G)\). A resolving \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\) is said to be connected if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is connected in \(G\). The minimum \(k\) for which there is a connected resolving \(k\)-partition of \(V(G)\) is called the connected partition dimension of \(G\), denoted by \(cpd(G)\). In this paper, the partition dimension as well as the connected partition dimension of the wheel \(W_n\) with \(n\) spokes are considered, by showing that \(\lceil (2n)^{1/3} \rceil \leq pd(W_n) \leq \lceil 2(n)^{1/2} \rceil +1\) and \(cpd(W_n) = \lceil (n+2)/3 \rceil\) for \(n \geq 4\).

Vertices \(x\) and \(y\) are called paired in tournament \(T\) if there exists a vertex \(z\) in the vertex set of \(T\) such that either \(x\) and \(y\) beat \(z\) or \(z\) beats \(x\) and \(y\). Vertices \(x\) and \(y\) are said to be distinguished in \(T\) if there exists a vertex \(z\) in \(T\) such that either \(x\) beats \(z\) and \(z\) beats \(y\), or \(y\) beats \(z\) and \(z\) beats \(x\). Two vertices are strictly paired (distinguished) in \(T\) if all vertices of \(T\) pair (distinguish) the two vertices in question. The \(p/d\)-graph of a tournament \(T\) is a graph which depicts strictly paired or strictly distinguished pairs of vertices in \(T\). \(P/d\)-graphs are useful in obtaining the characterization of such graphs as domination and domination-compliance graphs of tournaments. We shall see that \(p/d\)-graphs of tournaments have an interestingly limited structure as we characterize them in this paper. In so doing, we find a method of constructing a tournament with a given \(p/d\)-graph using adjacency matrices of tournaments.

Let \(G\) be a simple connected graph. The spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue of its adjacency matrix. In this paper, we obtain two lower bounds of \(\rho(G)\) by two different methods, one of which is better than another in some conditions.

In this note we compute the chromatic polynomial of the Jahangir graph \(J_{2p}\) and we prove that it is chromatically unique for \(p=3\).

In this paper, we compute the PI and Szeged indices of some important classes of benzenoid graphs, which some of them are related to nanostructures. Some open questions are also included.

The detour \(d(i, j)\) between vertices \(i\) and \(j\) of a graph is the number of edges of the longest path connecting these vertices. The matrix whose \((i, j)\)-entry is the detour between vertices \(i\) and \(j\) is called the detour matrix. The half sum \(D\) of detours between all pairs of vertices (in a connected graph) is the detour index, i.e.,

\[D = (\frac{1}{2}) \sum\limits_j\sum\limits_i d(i,j)\]

In this paper, we computed the detour index of \(TUC_4C_8(S)\) nanotube.

We construct several new group divisible designs with block size five and with \(2, 3\), or \(6\) groups.

A list \((2,1)\)-labeling \(\mathcal{L}\) of graph \(G\) is an assignment list \(L(v)\) to each vertex \(v\) of \(G\) such that \(G\) has a \((2,1)\)-labeling \(f\) satisfying \(f(v) \in L(v)\) for all \(v\) of graph \(G\). If \(|L(v)| = k + 1\) for all \(v\) of \(G\), we say that \(G\) has a \(k\)-list \((2,1)\)-labeling. The minimum \(k\) taken over all \(k\)-list \((2,1)\)-labelings of \(G\), denoted \(\lambda_l(G)\), is called the list label-number of \(G\). In this paper, we study the upper bound of \(\lambda(G)\) of some planar graphs. It is proved that \(\lambda_l(G) \leq \Delta(G) + 6\) if \(G\) is an outerplanar graph or \(A\)-graph; and \(\lambda_l(G) \leq \Delta(G) + 9\) if \(G\) is an \(HA\)-graph or Halin graph.

In this paper, we give a necessary and sufficient condition for a \(3\)-regular graph to be cordial.