It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k-thickening or k-stretching, then its Tutte polynomial can be expressed in terms of the Tutte pol...

The total graph is built by joining the to its line means of incidences. We introduce a similar construction for signed graphs. Under two defnitions graph, we defne corresponding and show that it stable under switching. consider balance, frustration index number, largesteigenvalue. In regular case compute spectrum adjacency matrix spectra certain compositions, determine some with exactly main e...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

a signed graph (marked graph) is an ordered pair $s=(g,sigma)$$(s=(g,mu))$, where $g=(v,e)$ is a graph called the underlyinggraph of $s$ and $sigma:erightarrow{+,-}$$(mu:vrightarrow{+,-})$ is a function. for a graph $g$, $v(g),e(g)$ and $c(g)$ denote its vertex set, edge set and cut-vertexset, respectively. the lict graph $l_{c}(g)$ of a graph $g=(v,e)$is defined as the graph having vertex set ...

It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sum of nonnegative graph circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that indecomposable flows of graphs are those graph circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than that of...