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 \([1]\).

Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \(f\)-factor if \(d_G^{h}(x) = f(x)\) for every \(x \in V(F)\). In this paper, we prove that if \(\delta(G) \geq b\) and \(\alpha(G) \leq \frac{4a(\delta-b)}{(b+1)^2}\), then \(G\) has a fractional \(f\)-factor. Where \(a\) and \(b\) are integers such that \(0 \leq a \leq f(x) \leq b\) for every \(x \in V(G)\). Therefore, we prove that the fractional analogue of Conjecture in \([2]\) is true.

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group, \(g \in A\) and \(\Gamma\) a group of automorphisms of \(D\). We consider the number of \(T\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. Specifically, we enumerate the number of \(I\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order and the trivial automorphism group \(\Gamma\) of \(D\), when \(A\) is the cyclic group \({Z}_{p^n}\) and the direct sum of \(m\) copies of \({Z}_p\) for any prime number \(p (> 2)\).

The Grundy number of an impartial game \(G\) is the size of the unique Nim heap equal to \(G\). We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these “fractal sequences” and give a bijection between these sequences and certain upper-triangular arrays. As a special case, we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap.