Let \(G\) be a connected graph. A weakly connected dominating set of \(G\) is a dominating set \(D\) such that the edges not incident to any vertex in \(D\) do not separate the graph \(G\). In this paper, we first consider the relationship between weakly connected domination number \(\gamma_w(G)\) and the irredundance number \(ir(G)\). We prove that \(\gamma_w(G) \leq \frac{5}{2}ir(G) – 2\) and this bound is sharp. Furthermore, for a tree \(T\), we give a sufficient and necessary condition for \(\gamma_c(T) = \gamma_w(T) + k\), where \(\gamma_c(T)\) is the connected domination number and \(0 \leq k \leq \gamma_w(T) – 1\).

For two vertices \(u\) and \(v\) in a strong digraph \(D\), the strong distance \(sd(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). The strong eccentricity \(se(v)\) of a vertex \(v\) of \(D\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(srad(D)\) (resp. strong diameter \(sdiam(D)\)) of \(D\) is the minimum (resp. maximum) strong eccentricity among all vertices of \(D\). The lower (resp. upper) orientable strong radius \(srad(G)\) (resp. \(SRAD(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong radius over all strong orientations of \(G\). The lower (resp. upper) orientable strong diameter \(sdiam(G)\) (resp. \(SDIAM(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong diameter over all strong orientations of \(G\). In this paper, we determine the lower orientable strong radius and strong diameter of the Cartesian product of complete graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of the Cartesian product of complete graphs.

In this paper, we show that the disjoint union of two cordial graphs, one of them is of even size, is cordial and the join of two cordial graphs, both are of even size or one of them is of even size and one of them is of even order, is cordial. We also show that \(C_m \cup C_n \) is cordial if and only if \(m+n \not\equiv 2 \pmod{4}\) and \(mC_n\) is cordial if and only if \(mn \not\equiv 2 \pmod{4}\) and for \(m, n \geq 3\), \(C_m + C_n\) is cordial if and only if \((m, n) \neq (3, 3)\) and \(\{m, n\} \not\equiv \{0, 2\} \pmod{4}\).

Finally, we discuss the cordiality of \(P_n^k\).

We show that a finite linear space with \(b = n^2 + n + 1\) lines, \(n \geq 2\), constant point-degree \(n+1\) and containing a sufficient number of lines of size \(n\) can be embedded in a projective plane of order \(n\). Using this fact, we also give characterizations of some pseudo-complements, which are the complements of certain subsets of finite projective planes.

The numbers of distinct self-orthogonal Latin squares (SOLS) and idempotent SOLS have been enumerated for orders up to and including $9$. The isomorphism classes of idempotent SOLS have also been enumerated for these orders. However, the enumeration of the isomorphism classes of non-idempotent SOLS is still an open problem. By utilising the automorphism groups of class representatives from the already enumerated isomorphism classes of idempotent SOLS, we enumerate the isomorphism classes of non-idempotent SOLS implicitly (i.e. without generating them). New symmetry classes of SOLS are also introduced, based on the number of allowable transformations that may be applied to a SOLS without destroying the property of self-orthogonality, and these classes are also enumerated.

The supereulerian index of a graph \(G\) is the smallest integer \(k\) such that the \(k\)-th iterated line graph of \(G\) is supereulerian. We first show that adding an edge between two vertices with degree sums at least three in a graph cannot increase its supereulerian index. We use this result to prove that the supereulerian index of a graph \(G\) will not be changed after either of contracting an \(A_G(F)\)-contractible subgraph \(F\) of a graph \(G\) and performing the closure operation on \(G\) (if \(G\) is claw-free). Our results extend Catlin’s remarkable theorem \([4]\) relating that the supereulericity of a graph is stable under the contraction of a collapsible subgraph.

This paper is based on the splitting operation for binary matroids that was introduced by Raghunathan, Shikare and Waphare [ Discrete Math. \(184 (1998)\), p.\(267-271\)] as a natural generalization of the corresponding operation in graphs. Here, we consider the problem of determining precisely which graphs \(G\) have the property that the splitting operation, by every pair of edges, on the cycle matroid \(M(G)\) yields a graphic matroid. This problem is solved by proving that there are exactly four minor-minimal graphs that do not have this property.

In this paper, we study the connection of number theory with graph theory via investigating some uncharted properties of the directed graph \(\Gamma'(n)\) whose vertex set is \(\mathbb{Z}_n = \{0,1,\ldots,n-1\}\), and for which there is a directed edge from \(a \in \mathbb{Z}_n\) to \(b \in \mathbb{Z}_n\) if and only if \(a^3 \equiv b \pmod{n}\). For an arbitrary prime \(p\), the formula for the decomposition of the graph \(\Gamma(p)\) is established. We specify two subgraphs \(\Gamma_1(n)\) and \(\Gamma_2(n)\) of \(\Gamma(n)\). Let \(\Gamma_1(n)\) be induced by the vertices which are coprime to \(n\) and \(\Gamma_2(n)\) by induced by the set of vertices which are not coprime to \(n\). We determine the level of every component of \(\Gamma_1(n)\), and establish necessary and sufficient conditions when \(\Gamma_1(n)\) or \(\Gamma_2(n)\) has no cycles with length greater than \(1\), respectively. Moreover, the conditions for the semiregularity of \(\Gamma_2(n)\) are presented.

We made a computer search for minimal blocking sets in the projective geometry \(\text{PG}(2,11)\), and found \(30,000\), of which only two nontrivial blocking sets had the possibility of being isomorphic.

A graph \(G\) is called \({claw-free}\) if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). Ando et al. obtained the result: a claw-free graph \(G\) with minimum degree at least \(d\) has a path-factor such that the order of each path is at least \(d+1\); in particular \(G\) has a \(\{P_3, P_4, P_5\}\)-factor whenever \(d \geq 2\). Kawarabayashi et al. proved that every \(2\)-connected cubic graph has a \(\{P_3, P_4\}\)-factor. In this article, we show that if \(G\) is a connected claw-free graph with at least \(6\) vertices and minimum degree at least \(2\), then \(G\) has a \(\{P_3, P_4\}\)-factor. As an immediate consequence, it follows that every claw-free cubic graph (not necessarily connected) has a \(\{P_3, P_4\}\)-factor.