Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard). More than one guard is allowed to move in response to an attack. The \( m \)-eternal domination number is the minimum number of guards needed to defend the graph. We characterize the trees achieving several upper and lower bounds on the \( m \)-eternal domination number.

Introduced in 1947, the Wiener index (sum of distances between all pairs of vertices) is one of the most studied chemical indices. Extensive results regarding the extremal structure of the Wiener index exist in the literature. More recently, the Gamma index (also called the Terminal Wiener index) was introduced as the sum of all distances between pairs of leaves. It is known that these two indices coincide in their extremal structures and that a nice functional relation exists for \(k\)-ary trees but not in general. In this note, we consider two natural extensions of these concepts, namely the sum of all distances between internal vertices (the Spinal index) and the sum of all distances between internal vertices and leaves (the Bartlett index). We first provide a characterization of the extremal trees of the Spinal index under various constraints. Then, its relation with the Wiener index and Gamma index is studied. The functional relation for \(k\)-ary trees also implies a similar result on the Bartlett index.

For an \( n \)-connected graph \( G \), the \( n \)-wide diameter \( d_n(G) \) is the minimum integer \( m \) such that for any two vertices \( x \) and \( y \) there are at least \( n \) internally disjoint paths of length at most \( m \) from \( x \) to \( y \). For a given integer \( l \), a subset \( S \) of \( V(G) \) is called a \( (l,n) \)-dominating set of \( G \) if for any vertex \( x \in V(G) – S \) there are at least \( n \) internally disjoint paths of length at most \( l \) from \( S \) to \( x \). The minimum cardinality among all \( (l,n) \)-dominating sets of \( G \) is called the \( (l,n) \)-domination number. In this paper, we obtain that the \( (l,\omega) \)-domination numbers of the circulant digraph \( G(d^n; \{1, d, \ldots, d^{n-1}\}) \) is equal to 2 for \( 1 \leq \omega \leq n \) and \( d_\omega(G) – (g(d,n) + \delta) \leq l \leq d_\omega(G) – 1 \), where \( g(d,n) = \text{min} \{e\lceil \frac{n}{2} \rceil – e – 2, (\lfloor \frac{n}{2} \rfloor + 1)(e – 1) – 2\} \), \( \delta = 0 \) for \( 1 \leq \omega \leq n – 1 \) and \( \delta = 1 \) for \( \omega = n \).

It is known that there are at least 8784 non-isomorphic designs with parameters \(2-(64, 28, 12)\) whose derived \(2-(28, 12, 11)\) designs are quasi-symmetric. In this paper, we examine the binary codes related to a class of non-isomorphic designs with these parameters and invariant under the Frobenius group of order 21 for which the derived \(2-(28, 12, 11)\) designs are not quasi-symmetric. We show that up to equivalence, there are 30 non-isomorphic binary codes obtained from them. Moreover, we classify the self-orthogonal doubly-even codes of length 13 obtained from the non-fixed parts of orbit matrices of these \(2-(64, 28, 12)\) designs under an action of an automorphism group of order four having 12 fixed points. The subcodes of codimension 1 and minimum weight 8 in these codes are all optimal single weight codes.

Let \( N(n, t_1, \ldots, t_r) \) be the number of irreducible polynomials of degree \( n \) over the finite field \( \mathbb{F}_2 \) where the coefficients of the terms \( x^{n-1}, \ldots, x^{n-r} \) are prescribed. Finding the exact values for the numbers \( N(n, t_1, \ldots, t_r) \) for \( r \geq 4 \) seems difficult. In this paper, we give an approximation for these numbers. We treat in detail the case \( N(n, t_1, \ldots, t_4) \), and we state the approximation in the general case. We experimentally show how good our approximation is.

The degree sequence of a finite graph \( G \) is its list \( D = (d_1, \ldots, d_n) \) of vertex degrees in non-increasing order. The graph \( G \) is called a realization of \( D \). In this paper, we characterize the graphic degree sequences \( D \) such that no realization of \( D \) contains an induced four-cycle. Our characterization is stated in terms of the class of forcibly chordal graphs.

A dice family \( D(n, a, b, s) \) includes all lists \( (x_1, \ldots, x_n) \) of integers with \( n \geq 1 \), \( a \leq x_1 \leq \ldots \leq x_n \leq b \), and \( \sum x_i = s \). Given two dice \( X \) and \( Y \), we compare the number of pairs \( (i, j) \) with \( x_i y_j \). If the second number is larger, then \( X \) is \({stronger}\) than \( Y \), and if the two numbers are equal, then \( X \) and \( Y \) are \({tied}\). In previous work, it has been observed that the density of ties in \( D(n, a, b, s) \) is generally lower than one might expect. In this note, we provide more information about this observation by calculating the asymptotic proportion of ties in certain kinds of dice families. Many other properties of dice families remain to be determined.

After introducing and discussing the notion of length two path centered surface area for general graphs, particularly for bipartite graphs, we derive a closed-form expression and an explicit expression for the length two path centered surface areas of the hypercube and the star graph, respectively.

The Ramsey numbers \( r(F, G) \) are investigated, where \( F \) is a non-tree graph of order \( 5 \) and minimum degree \( 1 \), and \( G \) is a connected graph of order \( 6 \). For all pairs \( (F, G) \) where \( F \neq K_5 – K_{1,3} \), the exact value of \( r(F, G) \) is determined. In order to settle \( F = K_5 – K_{1,3} \), we prove \( r(K_5 – K_{1,3}, G) = r(K_4, G) \).

A Stanton-type graph \( S(n, m) \) is a connected multigraph on \( n \) vertices such that for a fixed \( m \) with \( n-1 \leq m \leq \binom{n}{2} \), there is exactly one edge of multiplicity \( i \) (and no others) for each \( i = 1, 2, \ldots, m \). In this note, we show how to decompose \( \lambda K_n \) (for the appropriate minimal values of \( \lambda \)) into Stanton-type graphs \( S(4, 3) \) of the LOE and OLE types.