We consider the placement of detection devices at the vertices of a graph \( G \), where a detection device at vertex \( v \) has three possible outputs: there is an intruder at \( v \); there is an intruder at one of the vertices in the open neighborhood \( N(v) \), the set of vertices adjacent to \( v \), but which vertex in \( N(v) \) cannot be determined; or there is no intruder in \( N[v] = N(v) \cup \{v\} \). We introduce the \( 1 \)-step locating-dominating problem of placing the minimum possible number of such detection devices in \( V(G) \) so that the presence of an intruder in \( V(G) \) can be detected, and the exact location of the intruder can be identified, either immediately or when the intruder has moved to an adjacent vertex. Some related problems are introduced.

Recently, four new vertex colorings of graphs (in which adjacent vertices may be colored the same) were introduced for the purpose of distinguishing every pair of adjacent vertices. For each graph and for each of these four colorings, the minimum number of required colors never exceeds the chromatic number of the graph. In this paper, we summarize some of the results obtained on these colorings and introduce some relationships among them.

We address the problem: for which values of \( d \) and \( n \) does there exist a triangle-free regular graph of degree \( d \) on \( n \) vertices? A complete solution is given.

Let \( G = K_{a,b} \), where \( a, b \) are even, or \( G = K_{a,a} – M_{2a} \), where \( a \geq 1 \) is an odd integer and \( M_{2a} \) is a perfect matching in \( K_{a,a} \). It has been shown ([3,4]) that \( G \) is arbitrarily decomposable into closed trails. Billington asked if the graph \( K_{r,s} – F \), where \( s, r \) are odd and \( F \) is a (smallest possible) spanning subgraph of odd degree, is arbitrarily decomposable into closed trails ([2]).

In this article we answer the question in the affirmative.

This paper considers the Lehmer matrix and its recursive analogue. The determinant of the Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations. We further define a generalized Lehmer matrix with \((i,j)\) entries \( g_{ij} = \frac{\text{min} \{u_{i+1}, u_{j+1}\}}{\text{max} \{u_{i+1}, u_{j+1}\}} \) where \( u_n \) is the \( n \)th term of a binary sequence \(\{u_n\}\). We derive both the LU and Cholesky factorizations of this analogous matrix and we precisely compute the determinant.

Random number generators are a small part of any computer simulation project. Yet they are the heart and the engine that drives the project. Often times software houses fail to understand the complexity involved in building a random number generator that will satisfy the project requirements and will be able to produce realistic results. Building a random number generator with a desirable periodicity, that is uniform, that produces all the random permutations with equal probability, and at random, is not an easy task. In this paper we provide tests and metrics for testing random number generators for uniformity and randomness. These tests are in addition to the already existing tests for uniformity and randomness, which we modify by running each test a large number of times on sub-sequences of random numbers, each of length \( n \). The test result obtained each time is used to determine the probability distribution function. This eliminates the random number generator misclassification error. We also provide new tests for uniformity and randomness, the new tests for uniformity test the skewness of each one of the subgroups as well as the kurtosis. The tests for randomness, which include the Fourier spectrum, the phase spectrum, the discrete cosine transform spectrum, and the orthogonal wavelet domain, test for patterns not detected in the raw data space. Finally we provide visual and acoustic tests.

For a connected graph \( G \) of order \( n \), the detour distance \( D(u, v) \) between two vertices \( u \) and \( v \) in \( G \) is the length of a longest \( u-v \) path in \( G \). A Hamiltonian labeling of \( G \) is a function \( c: V(G) \to \mathbb{N} \) such that

\[ |c(u) – c(v)| + D(u,v) \geq n \]

for every two distinct vertices \( u \) and \( v \) of \( G \). The value \( \text{hn}(c) \) of a Hamiltonian labeling \( c \) of \( G \) is the maximum label (functional value) assigned to a vertex of \( G \) by \( c \); while the Hamiltonian labeling number \( \text{hn}(G) \) of \( G \) is the minimum value of a Hamiltonian labeling of \( G \). We present several sharp upper and lower bounds for the Hamiltonian labeling number of a connected graph in terms of its order and other distance parameters.

A graph \( G \) is \( 3 \)-existentially closed (\( 3 \)-e.c.) if each \( 3 \)-set of vertices can be extended in all of the possible eight ways. Results which improve the lower bound of the minimum order of a \( 3 \)-e.c. graph are reported. It has been shown that \( m_{ec}(3) \geq 24 \), where \( m_{ec}(3) \) is defined to be the minimum order of a \( 3 \)-e.c. graph.

In this study, we analyze the structure of the full collineation group of certain Veblen-Wedderburn (VW) planes of orders \( 5^2 \), \( 7^2 \), and \( 11^2 \). We also discuss a reconstruction method using their collineation groups.