Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if G ( ) V G ( ) e E v ( ) E G G : ( f E k b k ) { 1,1} G   ( , ) b k ( ) f e b    for at least vertices of , where . The value k v G ( ) = {uv E( ( ) E v G u N v   ) | } ( ) min ( ) G e E f e   , taki...

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it ...

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ∈ {3, 4, 5, 6}, every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of le...

Let G be a simple graph without isolated vertex with vertex set V (G) and edge set E(G). A function f : E(G) −→ {−1, 1} is said to be a signed star k-subdominating function of G if ∑ e∈E(v) f (e) ≥ 1 for at least k vertices v of G, where E(v) = {uv ∈ E(G) | u ∈ N (v)}. The value min ∑ e∈E(G) f (e), taking over all signed star k-subdominating function f of G is called the signed star k-subdomina...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}, called the signature of S. The ×-line sigraph of S denoted by L×(S) is a sigraph defined on the line graph L(S u) of the graph Su by assigning to each edge ef of L(Su), the product of si...

Previous studies on social networks are often focused on networks with only positive relations between individual nodes. As a significant extension, we conduct the spectral analysis on graphs with both positive and negative edges. Specifically, we investigate the impacts of introducing negative edges and examine patterns in the spectral space of the graph’s adjacency matrix. Our theoretical res...

The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If ∑ x∈N [e] f(x) ≥ 1 for at least k edges e of G, then f is called a signed edge k-subdominating function of G. The minimum of the values ∑ e∈E(G) f(e), taken over all signed edge k-subdomina...

and The National Graduate School for Biomaterials and Tissue Engineering ACADEMIC DISSERTATION To be publicly discussed with the permission of the Medical Faculty of the University of Helsinki, in the large lecture hall (Iso luentosali) Work hard and try your best! If you then fail I am not angry.-my father-This thesis is based on the following original publications. They are referred to in the...