The Zagreb indices are topological indices of graphs, which are defined as:\(M_1(G) = \sum\limits_{v \in V(G)} (d(v))^2\), \(M_2(G) = \sum\limits_{uv \in E(G)} d(u)d(v)\) .In this paper, we determine the upper and lower bounds for the Zagreb indices of unicyclic graphs in terms of their order and girth. In each case, we characterize the extremal graphs.

In this paper, we are concerned with Leibniz numbers. We establish a series of identities involving Leibniz numbers, Stirling numbers, harmonic numbers, arctan numbers by making use of generating functions. In addition, we give the asymptotic expansion of certain sums related to Leibniz numbers by Laplace’s method.

We consider the undirected simple connected graph for which edges fail independently of each other with equal probability \(1 – p\) and nodes are perfect. The all-terminal reliability of a graph \(G\) is the probability that the spanning subgraph of surviving edges is connected, denoted as \(R(G,p)\). Graph \(G \in \Omega(n,e)\) is said to be uniformly least reliable if \(R(G,p) \leq R(G’,p)\) for all \(G’ \in \Omega(n,e)\), and for all edge failure probabilities \(0 < 1 – p < 1\). In this paper, we prove the existence of uniformly least reliable graphs in the class \(\Omega(n,e)\) for \(e \leq n + 1\) and give their topologies.

We study V- and \(\Lambda\)-patterns which generalize valleys and peaks, as well as increasing and decreasing runs, in permutations. A complete classification of permutations (multi)-avoiding V- and \(\Lambda\)-patterns of length \(4\) is given. We also establish a connection between restricted permutations and matchings in the coronas of complete graphs.

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.