Let \(\operatorname{MPT}(v,\lambda)\) denote a maximum packing of triples of order \(v\) with index \(\lambda\). For \(\lambda > 1\) and \(v \geq 3\), it is proved in this paper that the necessary and sufficient condition for the embedding of an \(\operatorname{MPT}(v,\lambda)\) in an \(\operatorname{MPT}(u,\lambda)\) is \(u \geq 20v + 1\).

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric \(Q(2n,2)\), \(n \geq 2\), which are not contained in a given hyperbolic quadric \(Q_+(2n-1,q) \subset Q(2n,q)\) define a sub near polygon \(\mathbb{I}_n\) of the dual polar space \(DQ(2n,2)\). It is known that every valuation of \(DQ(2n,2)\) induces a valuation of \(\mathbb{I}_n\). In this paper, we show that also the converse is true: every valuation of \(\mathbb{I}_n\) is induced by a valuation of \(DQ(2n,2)\). We will also study the structure of the valuations of \(\mathbb{I}_n\).

The (Laplacian) spectral radius of a graph is the maximum eigenvalue of its adjacency matrix (Laplacian matrix, respectively). Let \(\mathcal{G}(n,k)\) be the set of bipartite graphs with \(n\) vertices and \(k\) blocks. This paper gives a complete characterization for the extremal graph with the maximum spectral radius (Laplacian spectral radius, respectively) in \(\mathcal{G}(n, k)\).

In the paper “A note on the eigenvalues of graphs, Ars Combinatoria \(94 (2010), 221-227\)” by Lihua Feng and Guihai Yu, page 226, we have the following note.

In this paper, we show that among all connected graphs of order \(n\) with diameter \(D\), the graph \(G^*\) has maximal spectral radius, where \(G^*\) is obtained from \(K_{n-D} \bigvee \overline{K_2}\) by attaching two paths of order \(l_1\) and \(l_2\) to the two vertices \(u,v\) in \(\overline{K_2}\), respectively, and \(l_1 + l_2 = D-2\), \(|l_1 – l_2| \leq 1\).

P. Erdés and T. Gallai gave necessary and sufficient conditions for a sequence of non-negative integers to be graphic. Here,their result is generalized to multigraphs with a specified multiplicity. This both generalizes and provides a new proof of a result in the literature by Chungphaisan \([2].\)

Let \(u\) and \(v\) be two vertices in a graph \(G\). We say vertex \(u\) dominates vertex \(v\) if \(N(v) \subseteq N(u) \cup \{u\}\). If \(u\) dominates \(v\) or \(v\) dominates \(u\), then \(u\) and \(v\) are comparable. The Dilworth number of a graph \(G\), denoted \(\operatorname{Dil}(G)\), is the largest number of pairwise incomparable vertices in the graph \(G\). A graph \(G\) is called claw-free if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). It is shown that if \(G\) is a \(k\) (\(k \geq 3\)) – connected claw-free graph with \(\operatorname{Dil}(G) \leq 2k-5\), then \(G\) is Hamilton-connected and a Hamilton path between every two vertices in \(G\) can be found in polynomial time.

In this paper, we analyze the familiar straight insertion sort algorithm and quantify the deviation of the output from the correct sorted order if the outcomes of one or more comparisons are in error. The disarray in the output sequence is quantified by six measures. For input sequences whose length is large compared to the number of errors, a comparison is made between the robustness to errors of bubble sort and the robustness to errors of straight insertion sort. In addition to analyzing the behaviour of straight insertion sort, we review some inequalities among the various measures of disarray, and prove some new ones.

In this article, we give new lower bounds for the size of edge chromatic critical graphs with maximum degrees of \(8\) and \(9\), respectively. Furthermore, it implies that if \(G\) is a graph embeddable in a surface \(S\) with characteristics \(c(S) = -1\) or \(-2\), then \(G\) is class one if maximum degree \(\Delta \geq 8\) or \(9\), respectively.

While powers of the adjacency matrix of a finite graph reveal information about walks on the graph, they fail to distinguish closed walks from cycles. Using elements of an appropriate commutative, nilpotent-generated algebra, a “new” adjacency matrix \(\Lambda\) can be associated with a random graph on \(n\) vertices. Letting \(X_k\) denote the number of \(k\)-cycles occurring in a random graph, this algebra together with a probability mapping allow \(\mathbb{E}(X_k)\) to be recovered in terms of \(\operatorname{tr} \Lambda^k\). Higher moments of \(X_k\) can also be computed, and conditions are given for the existence of higher moments in growing sequences of random graphs by considering infinite-dimensional algebras. The algebras used can be embedded in algebras of fermion creation and annihilation operators, thereby establishing connections with quantum computing and quantum probability theory. In the framework of quantum probability, the nilpotent adjacency matrix of a finite graph is a quantum random variable whose \(m\)th moment corresponds to the \(m\)-cycles contained in the graph.