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...

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.

A signed graph (or sigraph in short) is an ordered pair $S=(S^u,\sigma )$, where $S^u$ is a graph G = (V,E), called the underlying graph of S and σ:E → +, - is a function from the edge set E of $S^u$ into the set +,-, called the signature of S. The ×-line sigraph of S denoted by $L_{\times }\left(S\right)$ is a sigraph defined on the line graph $L\left(S^u\right)$ of the graph $S^u$ by assigning to each edge ef of $L\left(S^u\right)$, the product of signs of the adjacent edges e and f in S. In this paper, first we define semi-total line sigraph and semi-total point sigraph of a given...