Related papers: Some New Results on Integer Additive Set-Valued Si…

The sets of vertices and edges of an undirected, simple, finite, connected graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. An arbitrary nonempty finite subset of consecutive integers is called an interval. An injective mapping…

Frank Harary introduced the concepts of sum and integral sum graphs. A graph $G$ is a \textit{sum graph} if the vertices of $G$ can be labeled with distinct positive integers so that $e = uv$ is an edge of $G$ if and only if the sum of the…

Let $S=(G, \sigma)$ be a signed graph where $G=(V, E)$ is a graph called the underlying graph of $S$ and $\sigma:E(G) \rightarrow \{+,~-\}$. Let $f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and…

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $…

A \emph{Fibonacci cordial labeling} of a graph \( G \) is an injective function \( f: V(G) \rightarrow \{F_0, F_1, \dots, F_n\} \), where \( F_i \) denotes the \( i^{\text{th}} \) Fibonacci number, such that the induced edge labeling \(…

Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to\mathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)\to\mathbb{Z}$ defined by…

A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss…

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…

Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,\sigma)$ is a weighted…

The automorphisms of a graph act naturally on its set of labeled imbeddings to produce its unlabeled imbeddings. The imbedding sum of a graph is a polynomial that contains useful information about a graph's labeled and unlabeled imbeddings.…

A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$…

We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite…

Acharya introduced the notion of set-valuations of graphs as a set analogue of the number valuations of graphs. Also we have the notion of set-indexers, integer additive set-indexers and k-uniform integer additive set-indexers. In this…

Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an…

In a signed graph, each link is labeled with either a positive or a negative sign. This is particularly appropriate to model polarized systems. Such a graph can be characterized through the notion of structural balance, which relies on the…

Let $\Sigma=(\Gamma, \sigma)$ is a signed graph(or sigraph in short), where $\Gamma$ is a underlying graph of $\Sigma$ and $\sigma:E\longrightarrow \{+, -\}$ is a function. Consider $\Gamma=Cay(\mathbb{Z}_{p_{1}}\times…

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ${\cal P}_1, >..., {\cal P}_n$ be additive hereditary graph properties. A graph $G$ has property $({\cal…

A signed graph $\Sigma$ is a pair $(G,\sigma)$, where $G=(V,E)$ is the underlying graph in which each edge is assigned $+1$ or $-1$ by the signature function $\sigma:E\rightarrow\{-1,+1\}$. In this paper, we extend the extensively applied…

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…

A \emph{signed graph} is a pair $\Gs$ in which $G$ is a finite simple graph and $\sigma:\E(G)\to\{+1,-1\}$ is a \emph{signature}. Following M\'a\v{c}ajov\'a--Raspaud- \v{S}koviera and Jin--Kang--Steffen, a \emph{proper coloring} of $\Gs$ is…