We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points.

The set $D$ of distinct signed degrees of the vertices in a signed graph $G$ is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.

Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If ${\sum }_{x\in N¯\left[v\right]}f\left(x\right)\ge 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number ${\gamma }_S\left(D\right)$ of D. A set $f₁,f₂,...,f_d$ of signed dominating functions on D with the property that ${\sum }_{i=1}^d{f}_i\left(x\right)\le 1$ for each...

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.

We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes.

Let k ≥ 2 be an integer. A function f: V(G) → −1, 1 defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence...

Let k ≥ 1 be an integer, and G = (V, E) be a finite and simple graph. The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and all edges having a common end-vertex with e. A signed Roman edge k-dominating function (SREkDF) on a graph G is a function f : E → {−1, 1, 2} satisfying the conditions that (i) for every edge e of G, ∑x∈NG[e] f(x) ≥ k and (ii) every edge e for which f(e) = −1 is adjacent to at least one edge e′ for which f(e′) = 2. The minimum of the values...

The signed total domination number of a graph is a certain variant of the domination number. If $v$ is a vertex of a graph $G$, then $N\left(v\right)$ is its oper neighbourhood, i.e. the set of all vertices adjacent to $v$ in $G$. A mapping $f:V\left(G\right)\to \{-1,1\}$, where $V\left(G\right)$ is the vertex set of $G$, is called a signed total dominating function (STDF) on $G$, if ${\sum }_{x\in N\left(v\right)}f\left(x\right)\ge 1$ for each $v\in V\left(G\right)$. The minimum of values ${\sum }_{x\in V\left(G\right)}f\left(x\right)$, taken over all STDF’s of $G$, is called the signed total domination number of $G$ and denoted by ${\gamma }_{\mathrm{s}t}\left(G\right)$. A theorem stating lower bounds for ${\gamma }_{\mathrm{s}t}\left(G\right)$ is stated for the...

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman dominating function (STRDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight of an STRDF f is w(f) = ∑v∈V (D) f(v). The signed total Roman domination number γstR(D) of D is the...

By a ternary system we mean an ordered pair $(W,R)$, where $W$ is a finite nonempty set and $R\subseteq W\times W\times W$. By a signpost system we mean a ternary system $(W,R)$ satisfying the following conditions for all $x,y,z\in W$: if $(x,y,z)\in R$, then $(y,x,x)\in R$ and $(y,x,z)\notin R$; if $x\ne y$, then there exists $t\in W$ such that $(x,t,y)\in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G,T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G,T)$ is a ct-pair, then by the guide to...

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].