Related papers: A Note on the Modified Albertson Index

The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) = \frac{1}{2}\sum\limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total…

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i(e)=|d(u)-d(v)|$, where $d(\cdot)$ is the vertex degree. The irregularity $I(G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end…

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1…

For a connected graph $G$, an instance $I$ is a set of pairs of vertices and a corresponding routing $R$ is a set of paths specified for all vertex-pairs in $I$. Let $\mathfrak{R}_I$ be the collection of all routings with respect to $I$.…

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 finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is…

A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where $\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in…

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound.…

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$. Let $d_i$ denote the degree of vertex $v_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times n}$, where $m_{ij}=\sqrt{(d_i + d_j -2)/(d_i d_j)}$ if $v_i…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017)…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

Very recently, the first geometric-arithmetic index $GA$ and arithmetic-geometric index $AG$ were introduced in mathematical chemistry. In the present paper, we first obtain some lower and upper bounds on $AG$ and characterize the extremal…

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper,…

A signed tree-coloring of a signed graph $(G,\sigma)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,\sigma)$ whose vertex set is the set of…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

A graph automorphism is a bijective mapping of the vertices that preserves adjacent vertices. A vertex determining set of a graph is a set of vertices such that the only automorphism that fixes those vertices is the identity. The size of a…