Related papers: Total Edge Irregularity Strength for Graphs

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$.…

We investigate the \textit{group irregularity strength}, $s_g(G)$, of a graph, i.e. the least integer $k$ such that taking any Abelian group $\mathcal{G}$ of order $k$, there exists a function $f:E(G)\rightarrow \mathcal{G}$ so that the…

Inspired by graceful labelings and total labelings of graphs, we introduce the idea of total difference labelings. A $k$-total labeling of a graph $G$ is an assignment of $k$ distinct labels to the edges and vertices of a graph so that…

A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident…

A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\cup E \to\{1,\ldots ,|V|+|E|\}$ such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights…

We introduce and study a generalization of conformal rigidity for graphs. A graph is $k$-conformally rigid if the uniform edge weights simultaneously maximize the sum of the $k$ smallest nontrivial Laplacian eigenvalues and minimize the sum…

Let $G$ be a finite simple undirected $(p,q)$-graph, with vertex set $V(G)$ and edge set $E(G)$ such that $p=|V(G)|$ and $q=|E(G)|$. A super edge-magic total labeling $f$ of $G$ is a bijection $f\colon V(G)\cup E(G)\longrightarrow…

We present a unified approach to compute the total vertex irregularity strength (tvs) of various graphs, employing a novel technique recently proposed by Barra et al. For graphs such as cycles, paths, prisms, wheels, complete graphs, helm…

For a graph $G$ the imbalance of an edge $uv$ of $G$ is $|deg_G(u)-deg_G(v)|$. Irregularity of a graph $G$ is defined as the sum of imbalances over all edges of $G$. In this paper we consider expansions and Pell graphs. If $H$ is an…

We investigate the \textit{group irregular strength} $(s_g(G))$ of graphs, i.e the smallest value of $s$ such that for any Abelian group $\Gamma$ of order $s$ exists a function $g\colon E(G) \rightarrow \Gamma$ such that sums of edge labels…

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup…

A vertex $v$ is called an AR-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. i.e., if $\{x_1,x_2,\dots,x_k\}$ are the edge labels of the edges incident on $v$, then the $2^k$ subset sums are…

Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\left\lfloor \frac{\Delta m}{\Delta n-m}\right\rfloor$, we…

The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let…

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give $G$ a strong edge-coloring is called the strong…

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen…

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. Obviously, ${\rm…

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the…

We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph…