The adjacent vertex distinguishing total chromatic number \(\chi_{at}(G)\) of a graph \(G\) is the smallest integer \(k\) for which \(G\) admits a proper \(k\)-total coloring such that no pair of adjacent vertices are incident to the same set of colors. Snarks are connected bridgeless cubic graphs with chromatic index \(4\). In this paper, we show that \(\chi_{at}(G) = 5\) for two infinite subfamilies of snarks, i.e., the Loupekhine snark and Blanusa snark of first and second kind. In addition, we give an adjacent vertex distinguishing total coloring using \(5\) colors for Watkins snark and Szekeres snark, respectively.

Let \(G\) be a tricyclic graph. Tricyclic graphs are connected graphs in which the number of edges equals the number of vertices plus two. In this paper, we determine graphs with the largest signless Laplacian spectral radius among all the tricyclic graphs with \(n\) vertices and diameter \(d\).

A pebbling move on a graph \(G\) consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph \(G\), denoted by \(f(G)\), is the least integer \(n\) such that, however \(n\) pebbles are located on the vertices of \(G\), we can move one pebble to any vertex by a sequence of pebbling moves. For any connected graphs \(G\) and \(H\), Graham conjectured that \(f(G \times H) \leq f(G)f(H)\). In this paper, we give the pebbling number of some graphs and prove that Graham’s conjecture holds for the middle graphs of some even cycles.

Graph embedding is an important factor to evaluate the quality of an interconnection network. It is also a powerful tool for implementation of parallel algorithms and simulation of different interconnection networks. In this paper, we compute the exact wirelength of embedding circulant networks into cycle-of-ladders.

In this paper, we characterize the extremal digraph with the maximal signless Laplacian spectral radius and the minimal distance signless Laplacian spectral radius among all simple connected digraphs with a given dichromatic number, respectively.

Given a graph \(G = (V, E)\) with no isolated vertex, a subset \(S \subseteq V\) is a total dominating set of \(G\) if every vertex in \(V\) is adjacent to a vertex in \(S\). A total dominating set \(S\) of \(G\) is a locating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\), we have \(N(u) \cap S \neq N(v) \cap S\), and \(S\) is a differentiating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V\), we have \(N(u) \cap S \neq N(v) \cap S\). The locating-total domination number (or the differentiating-total domination number) of \(G\), denoted by \(\gamma_t^L(G)\) (or \(\gamma_t^D(G)\)), is the minimum cardinality of a locating-total dominating set (or a differentiating-total dominating set) of \(G\). In this paper, we investigate the bounds of locating and differentiating-total domination numbers of unicyclic graphs.

Motzkin posed the problem of finding the maximal density \(\mu(M)\) of sets of integers in which the differences given by a set \(M\) do not occur. The problem is already settled when \(|M| \leq 2\) or \(M\) is a finite arithmetic progression. In this paper, we determine \(\mu(M)\) when \(M\) has some other structure. For example, we determine \(\mu(M)\) when \(M\) is a finite geometric progression.

For vertices \(u, v\) in a connected graph \(G\), a \(u-v\) chordless path in \(G\) is a \(u-v\) monophonic path. The monophonic interval \(J_G[u, v]\) consists of all vertices lying on some \(u-v\) monophonic path in \(G\). For \(S \subseteq V(G)\), the set \(J_G[S]\) is the union of all sets \(J_G[u, v]\) for \(u, v \in S\). A set \(S \subseteq V(G)\) is a monophonic set of \(G\) if \(J_G[S] = V(G)\). The cardinality of a minimum monophonic set of \(G\) is the monophonic number of \(G\), denoted by \(mn(G)\). In this paper, bounds for the monophonic number of the strong product graphs are obtained, and for several classes, improved bounds and exact values are obtained.

A hypergraph is a useful tool to model complex systems and can be considered a natural generalization of graphs. In this paper, we define some operations of fuzzy hypergraphs and strong fuzzy \(r\)-uniform hypergraphs, such as Cartesian product, strong product, normal product, lexicographic product, union, and join. We prove that if a hypergraph \(H\) is formed by one of these operations, then this hypergraph is a fuzzy hypergraph or a strong fuzzy \(r\)-uniform hypergraph. Finally, we discuss an application of fuzzy hypergraphs.

Let \(p_e(n)\) be the number of ways to make change for \(n\) cents using pennies, nickels, dimes, and quarters. By manipulating the generating function for \(p_e(n)\), we prove that the sequence \(\{p_e(n) \pmod{\ell^j}\}\) is periodic for every prime power \(\ell\).