It is shown that the circuit polynomial characterizes many of the well-known families of graphs. These include chains, stars, cycles, complete graphs, regular complete bipartite graphs, and wheels. Some analogous results are deduced for the characteristic polynomial and the \(\mu\)-polynomial.

A connected dominating set is a dominating set \(S\) with the additional property that the subgraph induced by \(S\) is connected. We are interested in the collection C of graphs in which every minimal connected dominating set is of one size.Trees, for instance, clearly belong to this collection. A partial characterization will be discussed; in particular, we determine those graphs which have the property that all spanning trees have the same number of leaves. It is noted that membership in this sub-collection of C can be determined in polynomial time.

A continuum with finitely many non-cut points is an irreducible tree. A two-variable power series for the number of (unlabelled) irreducible trees with \(p\) pendant and \(q\) interior vertices.The result is then specialized to get Harary’s series for the number of irreducible trees with \(n\) vertices and to another series for the number of irreducible trees with \(p\) pendant vertices, a result of interest in continuum theory.

The theory of lifting voltage digraphs provides a useful tool for constructing large digraphs with specified properties from suitable small base digraphs endowed with an assignment of voltages (= elements of a finite group) on arcs.
We revisit the degree/diameter problem for digraphs from this new perspective and prove a general upper bound on the diameter of a lifted digraph in terms of properties of the base digraph and voltage assignment.
In addition, we demonstrate that all currently known largest vertex-transitive Cayley digraphs for semidirect products of groups can be described by means of a voltage assignment construction using simpler groups.

The “characteristic” of a graph—the number of vertices, minus the number of edges, plus the number of triangles, etc.—is a little-studied, overtly combinatorial graph parameter intrinsically related to chordal graphs and common neighborhoods of subgraphs. I also introduce a sequence of related “higher characteristic” parameters.

A \({least \;deviant\; path}\) between two vertices in a weighted graph is defined as a path that minimizes the difference between the largest and smallest edge weights on the path.Algorithms are presented to determine the least deviant path. The fastest algorithm runs in \(O(|E|^{1.793})\), in the worst case. A type of two-dimensional binary search is used to achieve this running time.

An SOLS (self-orthogonal Latin square) of order \(n\) with \(n_i\) missing sub-SOLS (holes) of order \(h_i\) (\(1 \leq i \leq k\)), which are disjoint and spanning (i.e., \(\sum_{i=1}^{k} n_ih_i = n\)), is called a frame SOLS and denoted by \(\text{FSOLS}(h_1^{n_1}, h_2^{n_2}, \ldots, h_k^{n_k})\).In this article, it is shown that an \(\text{FSOLS}(3^{n-u}3^1)\) exists if and only if \(n \geq 4\) and \(n \geq 1 + \frac{2u}{3}\), with seventeen possible exceptions \((n, u) =\{(5, 1)\}\) and \(\{(n, u) = (n, \lfloor \frac{3(n-1)}{2}\rfloor)\) for \((n \in \{6, 10, 14, 18, 22, 30, 34, 38, 42, 46, 54, 58, 62, 66, 70, 94\} \).

It is straightforward to show that the full automorphism group of \(G \otimes K_n\) contains the Cartesian cross product of \(\text{Aut}(G)\) and \(S_n\). If \(\text{Aut}(G \otimes K_n)\) properly contains this cross product, then we will say that \(G \otimes K_n\) has a “rich” automorphism group. First, several conditions on \(G\) that ensure that \(G \otimes K_n\) has a rich automorphism group are given. Then, it is shown that these conditions are both necessary and sufficient for \(G \otimes K_n\) to have a rich automorphism group.

In this note, necessary and sufficient conditions are given for the existence of an equitable partial Steiner triple system \((S,T)\) on \(n\) symbols with exactly \(t\) triples, such that the leave of \((S,T)\) contains a \(1\)-factor if \(n\) is even and a near \(1\)-factor if \(n\) is odd.

A catalogue is presented which contains the graphs having order at most 10 which are critical with respect to the total chromatic number. A number of structural properties which cause these graphs to be critical are discussed, and a number of infinite classes of critical graphs are identified.A total colouring of a graph \(G\) is a function assigning colours to the vertices and edges of \(G\) in such a way that no two adjacent or incident elements are assigned the same colour. The total chromatic number, \(\chi”(G)\), is the minimum number of colours which need to be assigned to obtain a total colouring of the graph \(G\).
A longstanding conjecture, made independently by Behzad [3] and Vizing [17], claims that
\[
\Delta(G) + 1 \leq \chi”(G) \leq \Delta(G) + 2
\]
where \(\Delta(G)\) is the maximum degree of \(G\). The lower bound is sharp, the upper bound remains to be proved. A graph \(G\) is said to be Type 1 if \(\chi”(G) = \Delta(G) + 1\) and is said to be Type 2 if \(\chi”(G) \geq \Delta(G) + 2\).

We define a graph \(G\) to be critical with respect to the total chromatic number if \(G\) is connected and \(\chi”(G – e) < \chi''(G)\) for every edge \(e\) in \(G\). In Section 1 of this paper we identify all small order critical graphs, the catalogue of graphs is presented as a table of diagrams. In Section 2 we study structural properties of these graphs in order to identify features which cause a graph to be Type 2.