We introduce a generalisation of the traditional magic square, which proves useful in the construction of magic labelings of graphs. An order \(n\) sparse semi-magic square is an \(n \times n\) array containing the entries \(1, 2, \ldots, m\) (for some \(m < n^2\)) once each with the remainder of its entries \(0\), and its rows and columns have a constant sum \(k\). We discover some of the basic properties of such arrays and provide constructions for squares of all orders \(n \geq 3\). We also show how these arrays can be used to produce vertex-magic labelings for certain families of graphs.

A graph \(G\) on \(n\) vertices has a prime labeling if its vertices can be assigned the distinct labels \(1, 2, \ldots, n\) such that for every edge \(xy\) in \(G\), the labels of \(x\) and \(y\) are relatively prime. In this paper, we show that generalized books and \(C_m\) snakes all have prime labelings. In the process, we demonstrate a way to build new prime graphs from old ones.

In this paper, we studied that a linear space, which is the complement of a linear space having points are not on a trilateral or a quadrilateral in a projective subplane of order \(m\), is embeddable in a unique way in a projective plane of order \(n\). In addition, we showed that this linear space is the complement of certain regular hyperbolic plane in the sense of Graves \([5]\) with respect to a finite projective plane.

We give a combinatorial proof of Wilson’s Theorem: \(p\) divides \(\{(p – 1)! +1\}\) if \(p\) is prime.

The Padmakar-Ivan (PI) index of a graph \(G\) is defined as \(PI(G) = \sum[n_{eu} (e|G) + n_{ev}(e|G)]\) where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\) than to \(v\), \(n_{ev}(e|G)\) is the number of edges of \(G\) lying closer to \(v\) than to \(u\), and the summation goes over all edges of \(G\). The PI Index is a Szeged-like topological index developed very recently. In this paper, an exact expression for the PI index of the armchair polyhex nanotubes is given.

A finite planar set is \(k\)-isosceles for \(k \geq 3\), if every \(k\)-point subset of the set contains a point equidistant from the other two. This paper gives a \(4\)-isosceles set consisting of \(7\) points with no three on a line and no four on a circle.

For a group \(T\) and a subset \(S\) of \(T\), the bi-Cayley graph \(\text{BCay}(T, S)\) of \(T\) with respect to \(S\) is the bipartite graph with vertex set \(T \times \{0, 1\}\) and edge set \(\{\{(g, 0), (ag, 1)\} | g \in T, s \in S\}\). In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infinite families of cubic semisymmetric graphs.

We study the notion of path-congruence \(\Phi: T_1 \rightarrow T_2\) between two trees \(T_1\) and \(T_2\). We introduce the concept of the trunk of a tree, and prove that, for any tree \(T\), the trunk and the periphery of \(T\) are stable. We then give conditions for which the center of \(T\) is stable. One such condition is that the central vertices have degree \(2\). Also, the center is stable when the diameter of \(T\) is less than \(8\).

We call a cycle whose length is at most \(5\) a short cycle. In this paper, we consider the packing of short cycles in a graph with specified edges. A minimum degree condition is obtained, which is slightly weaker than that of the result in \([1]\).