For a vertex \(v\) of a connected graph \(G\) and a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v, S) = \min\{d(v,x) : x \in S\}\), where \(d(v,x)\) is the distance between \(v\) and \(x\). For an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the code of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(c_\Pi(v) = (d(v,S_1), d(v,S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is a resolving partition if the codes \(c_\Pi(v)\), \(v \in V(G)\), are distinct. A resolving partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) is acyclic if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is acyclic in \(G\). The minimum \(k\) for which there is a resolving acyclic \(k\)-partition of \(V(G)\) is the resolving acyclic number \(a_r(G)\) of \(G\). We study connected graphs with prescribed order, diameter, vertex-arboricity, and resolving acyclic number. It is shown that, for each triple \(d,k,n\) of integers with \(2 \leq d \leq n-2\) and \(3 \leq (n-d+1)/2 \leq k \leq n-d+1\), there exists a connected graph of order \(n\) having diameter \(d\) and resolving acyclic number \(k\). Also, for each pair \(a, b\) of integers with \(2 \leq a \leq b-1\), there exists a connected graph with resolving acyclic number \(a\) and vertex-arboricity \(b\). We present a sharp lower bound for the resolving acyclic number of a connected graph in terms of its clique number. The resolving acyclic number of the Cartesian product \(H \times K_2\) of nontrivial connected graph \(H\) and \(K_2\) is studied.

In this paper, we completely solve the problem of finding a maximum packing of any balanced complete multipartite graph \(K_{m}(n)\) with edge-disjoint \(6\)-cycles, and minimum leaves are explicitly given.

Subsequently, we also find a minimum covering of \(K_{m}(n)\).

Orthogonal designs and their special cases, such as weighing matrices and Hadamard matrices, have many applications in combinatorics, statistics, and coding theory, as well as in signal processing. In this paper, we generalize the definition of orthogonal designs, give many constructions for these designs, and prove some multiplication theorems that, most of them, can also be applied in the special case of orthogonal designs. Some necessary conditions for the existence of generalized orthogonal designs are also given.

We prove that the corona graphs \(C_n \circ K_1\) are \(k\)-equitable, as per Cahit’s definition of \(k\)-equitability, for \(k = 2, 3, 4, 5, 6\).

For a vertex \(v\) of a graph \(G = (V, E)\), the domination number \(\gamma(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a dominating set in \(G\) that contains \(v\). The average domination number of \(G\) is \(\gamma_{av}(G) = \frac{1}{|V|} \sum_{v\in V} \gamma_v(G)\). The independent domination number \(i_v(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a maximal independent set in \(G\) that contains \(v\). The average independent domination number of \(G\) is \(\gamma_{av}^i(G) = \frac{1}{|V|} \sum_{v\in V} i_v(G)\). In this paper, we show that a tree \(T\) satisfies \(\gamma_{av}(T) = i_{av}(T)\) if and only if \(A(T) = \vartheta\) or each vertex of \(A(T)\) has degree \(2\) in \(T\), where \(A(T)\) is the set of vertices of \(T\) that are contained in all its minimum dominating sets.

A graph \(G\) is \(K_r\)-covered if each vertex of \(G\) is contained in a clique \(K_r\). Let \(\gamma(G)\) and \(\gamma_t(G)\) respectively denote the domination and the total domination number of \(G\). We prove the following results for any graph \(G\) of order \(n\):

If \(G\) is \(K_6\)-covered, then \(\gamma_t(G) \leq \frac{n}{3}\),

If \(G\) is \(K_r\)-covered with \(r = 3\) or \(4\) and has no component isomorphic to \(K_r\), then \(\gamma_t(G) \leq \frac{2n}{r+1}\),

If \(G\) is \(K_3\)-covered and has no component isomorphic to \(K_3\), then \(\gamma(G) + \gamma_t(G) \leq \frac{7n}{9}\).

Corollaries of the last two results are that every claw-free graph of order \(n\) and minimum degree at least \(3\) satisfies \(\gamma_t(G) \leq \frac{n}{2}\) and \(\gamma(G) + \gamma(G) \leq \frac{7n}{9}\). For general values of \(r\), we give conjectures which would generalise the previous results. They are inspired by conjectures of Henning and Swart related to less classical parameters \(\gamma_{K_r}\) and \(\gamma^t_{K_r}\).

We are interested in linear-fractional transformations \(y,t\) satisfying the relations \(y^6=t^6 = 1\), with a view to studying an action of the subgroup \(H = \) on \({Q}(\sqrt{n}) \cup \{\infty\}\) by using coset diagrams.

For a fixed non-square positive integer \(n\), if an element \(\alpha = \frac{a+\sqrt {n}}{c}\) and its algebraic conjugate have different signs, then \(\alpha\) is called an ambiguous number. They play an important role in the study of action of the group \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\). In the action of \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\), \(\mathrm{Stab}_\alpha{(H)}\) are the only non-trivial stabilizers and in the orbit \(\alpha H\); there is only one (up to isomorphism). We classify all the ambiguous numbers in the orbit and use this information to see whether the action is transitive or not.

We are studying clique graphs of planar graphs, \(K(\text{Planar})\), this means the graphs which are the intersection of the clique family of some planar graph. In this paper, we characterize the \(K_3\) – free and \(K_4\) – free graphs which are in \(K(\text{Planar})\).