Let \(G\) be a group of permutations acting on an \(7\)-vertex set \(V\), and \(X\) and \(Y\) be two simple graphs on \(V\). We say that \(X\) and \(Y\) are \(G\)-isomorphic if \(Y\) belongs to the orbit of \(X\) under the action of \(G\). One can naturally generalize the reconstruction problems so that when \(G\) is \(S_v\), the symmetric group, we have the usual reconstruction problems. In this paper, we study \(G\)-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.

Frucht and Salinas [1] conjectured that \(C(k) \cup P(n)\) (\(n \geq 3\)) is graceful if and only if \(k + n \geq 7\). We prove that \(C(2k) \cup P(n)\) is graceful for \(n > k + 1\) (\(k \geq 3\)).

For smaller cases we prove that \(C(2k) \cup P(n)\) is graceful for \(k = 3, 4, 5, 6; n \geq 2\).

We present necessary and sufficient conditions for the decomposition of the complete symmetric bipartite digraph into each of the orientations of a \(4\)-cycle (in the cases for which such decompositions are not already known). We use these results to find optimal packings of the complete symmetric digraph with each of the orientations of a \(4\)-cycle. Finally, we give necessary and sufficient conditions for the existence of a decomposition of the complete symmetric digraph on \(v\) vertices with a hole of size \(w\) into each of the orientations of a \(4\)-cycle.

The sum graph of a set \(S\) of positive integers is the graph \(G^+(S)\) having \(S\) as its vertex set, with two distinct vertices adjacent whenever their sum is in \(S\). If \(S\) is allowed to be a subset of all integers, the graph so obtained is called an integral sum graph. The integral sum number of a given graph \(G\) is the smallest number of isolated vertices which when added to \(G\) result in an integral sum graph. In this paper, we find the integral sum numbers of caterpillars, cycles, wheels, and complete bipartite graphs.

Let \(k\) Max MOLS\((n)\) denote a maximal set of \(k\) mutually orthogonal Latin squares of order \(n\), and let the parameter triple \((G,n,k)\) denote the existence of a \(k\) Max MOLS\((n)\) constructed from orthogonal orthomorphisms of a group \(G\) of order \(x\). We identify all such parameter triples for all \(G\) of order \(\leq 15\), and report the existence of \(3\) Max MOLS\((n)\) for \(n = 15, 16\) and \(4\) Max MOLS\((n)\) for \(n = 12, 16, 24, 28\). Our work shows that for \(n \leq 15\), all known parameter pairs \((n, k)\) for which there exists a \(k\) Max MOLS\((n)\) can be attained by constructing maximal sets of MOLS from orthomorphisms of groups, except for \(1\) Max MOLS\((n)\), \(n = 5, 7, 9, 13\) and \(2\) Max MOLS\((10)\).

An alternating circular list of distinct \(r\)-element subsets of some finite set \(X\) and distinct \(r\)-partitions of type \(\tau\) is said to be a \(\tau\)-loop if successive members of the list are orthogonal. We address the problem of finding complete \(\tau\)-loops including all \(r\)-element subsets of \(X\), for any fixed \(|X|\) and type \(\tau\).

The general Randić index \(w_\alpha(G)\) of a graph \(G\) is the sum of the weights \(( d_G(u) d_G(v))^\alpha\) of all edges \(uv\) of \(G\). We give bounds for \(w_{-1}(T)\) when \(T\) is a tree of order \(n\). We also show that \(lim_{n\to\infty} f(n)/n\) exists, and give bounds for the limit, where \(f(n) = \max\{w_{-1}(T): T\) is a tree of order \(n\)}. Finally, we find the expected value and variance of \(w_\alpha(T)\) for certain families of trees.

The Hermitean forms graphs Her\((n,s)\) are a series of linear distance-regular graphs. The graph Her\((2,3)\) has the coset graph of the shortened ternary Golay code as an antipodal distance-regular cover. We give a new construction for this linear \(3\)-cover of \(Her\((2,3)\) and show that it is unique.

A cyclic triple, \((a, b, c)\), is defined to be the set \(\{(a, b), (b, c), (c, a)\}\) of ordered pairs. A Mendelsohn triple system of order \(v\), MTS\((v)\), is a pair \((M, \beta)\), where \(M\) is a set of \(v\) points and \(\beta\) is a collection of cyclic triples of pairwise distinct points of \(M\) such that any ordered pair of distinct points of \(M\) is contained in precisely one cyclic triple of \(\beta\). An antiautomorphism of a Mendelsohn triple system, \((M, \beta)\), is a permutation of \(M\) which maps \(\beta\) to \(\beta^{-1}\), where \(\beta^{-1} = \{(c, b, a) \mid (a, b, c) \in \beta\}\). In this paper, we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order \(v\) admitting an antiautomorphism consisting of two cycles of equal length and having \(0\) or \(1\) fixed points.

Let \(G\) be a finite group and let \(\nu_i(G)\) denote the proportion of ordered pairs of \(G\) that generate a subgroup of nilpotency class \(i\). Various properties of the \(\nu_i(G)\)’s are established. In particular, it is shown that \(\nu_i(G) = k_i |G|/|G|^2\) for some non-negative integer \(k_i\) and that \(\sum_{i=1}^{\infty}\nu_i\) is either \(1\) or at most \(\frac{1}{2}\) for solvable groups.