In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (\(LEE\)) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of \(LEE\)-graphs over the set of all connected spanning \(k\)-edge subgraphs of a connected graph \(G\), and show that both of them are connected for a \(2\)-connected graph \(G\).

We look at binary strings of length \(n\) which contain no odd run of zeros and express the total number of such strings, the number of zeros, the number of ones, the total number of runs, and the number of levels, rises, and drops as functions of the Fibonacci and Lucas numbers and also give their generating functions. Furthermore, we look at the decimal value of the sum of all binary strings of length \(n\) without odd runs of zeros considered as base \(2\) representations of decimal numbers, which interestingly enough are congruent (mod \(3\)) to either \(0\) or a particular Fibonacci number. We investigate the same questions for palindromic binary strings with no odd runs of zeros and obtain similar results, which generally have different forms for odd and even values of \(n\).

In this paper, we characterize the potentially \(K_4\)-graphic sequences. This characterization implies the value \(\sigma(K_4,n)\), which was conjectured by P. Erdős, M. S. Jacobson, and J. Lehel [1] and was confirmed by R. J. Gould, M. S. Jacobson, and J. Lehel [2] and Jiong-Sheng Li and Zixia Song [5], independently.

The graph …….is called a kite and the decomposition of \(K_n\) into kites is called a kite system. Such systems exist precisely when \(n = 0\) or \(1\) (mod \(8\)). In \(1975\), C. C. Lindner and A. Rosa solved the intersection problem for Steiner triple systems. The object of this paper is to give a complete solution to the triangle intersection problem for kite systems (\(=\) how many triangles can two kite systems of order \(n\) have in common). We show that if \(x \in \{0, 1, 2, \dots, n(n-1)/8\}\), then there exists a pair of kite systems of order \(n\) having exactly \(n\) triangles in common.

A directed balanced incomplete block design (\(DB\)(\(k\), \(\lambda\);\(v\))) \((X, \mathcal{B})\) is called self-converse if there is an isomorphic mapping \(f\) from \((X, \mathcal{B})\) to \((X, \mathcal{B}^{-1})\), where \(\mathcal{B}^{-1} = \{B^{-1} : B \in \mathcal{B}\}\) and \(B^{-1} = (x_k,x_{k-1},\ldots,x_2,x_1)\) for \(B=(x_1,x_2,\ldots,x_{k-1},x_{k})\). In this paper, we give the existence spectrum for self-converse \(DB\)(\(4\),\(\lambda\);\(v\)) for any \(\lambda \geq 1\).

A sequence \(\pi = (d_1, \dots, d_n)\) of nonnegative integers is graphic if there exists a graph \(G\) with \(n\) vertices for which \(d_1, \dots, d_n\) are the degrees of its vertices. \(G\) is referred to as a realization of \(\pi\). Let \(P\) be a graph property. A graphic sequence \(\pi\) is potentially \(P\)-graphic if there exists a realization of \(\pi\) with the graph property \(P\). Similarly, \(\pi\) is forcibly \(P\)-graphic if all realizations of \(\pi\) have the property \(P\). We characterize potentially Halin graph-graphic sequences, forcibly Halin graph-graphic sequences, and forcibly cograph-graphic sequences.

We establish the nonexistence of:(i) Steiner \(t\)-\((v,k)\) trades of volume \(s\), for \(2^t + 2^{t-1} < s t+1\) and volume \(s < (t-1)2^t + 2\).

Using R. C. Read’s superposition method, we establish a formula for the enumeration of Euler multigraphs, with loops allowed and with given numbers of edges. In addition, applying Burnside’s Lemma and our adaptation of Read’s superposition method, we also derive a formula for the enumeration of Euler multigraphs without loops — via the calculation of the number of perfect matchings of the complement of complete multipartite graphs. MAPLE is employed to implement these enumerations. For one up to \(13\) edges, the numbers of nonisomorphic Euler multigraphs with loops allowed are:\(1, 3, 6, 16, 34, 90, 213, 572, 1499, 4231, 12115, 36660, 114105\) respectively, and for one up to \(16\) edges, the numbers of nonisomorphic Euler multigraphs without loops are:\(0, 1, 1, 4, 4, 15, 22, 68, 131, 376, 892, 2627, 7217, 22349, 69271, 229553\) respectively. Simplification of these methods yields the numbers of multigraphs with given numbers of edges, results which also appear to be new. Our methods also apply to multigraphs with essentially arbitrary constraints on vertex degrees.

In this paper, we determine the number of all maximal \(k\)-independent sets in the generalized lexicographical product of graphs. We construct a polynomial that calculates this number using the concept of Fibonacci polynomials and generalized Fibonacci polynomials. Also, for special graphs, we give the recurrence formula.