Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

We introduce joins of signed graphs and explore the chromatic number all-positive all-negative joins. prove an analogue to theorem that join two equals sum their numbers. Given graphs, is usually less than numbers, by amount depends on new concept deficiency a signed-graph coloration.

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

A signed graph is a graph whose edges are signed. In a vertex-signed graph the vertices are signed. The latter is called consistent if the product of signs in every circle is positive. The line graph of a signed graph is naturally vertexsigned. Based on a characterization by Acharya, Acharya, and Sinha in 2009, we give constructions for the signed simple graphs whose naturally vertex-signed lin...

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

3 A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed domi4 nating function if for any vertex v the sum of function values over its closed neighborhood 5 is at least one. The signed domination number γs(G) of G is the minimum weight of a 6 signed dominating function on G. By simply changing “{+1,−1}” in the above definition 7 to “{+1, 0,−1}”, we can define the minus ...