We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph Kn is defined as 1 − n, and the rupture degree for an incomplete connected graph G is defined by r(G) = max{ω(G − X)− |X| − m(G − X) : X ⊂ V(G),ω(G − X) > 1}, where ω(G − X) is the number of components of G − X and m(G − X) is the order of a largest component of G − X . It is shown that this paramete...

A resolving set of a graph G is a set S ⊆ V (G), such that, every pair of distinct vertices of G is resolved by some vertex in S. The metric dimension of G, denoted by β(G), is the minimum cardinality of all the resolving sets of G. Shamir Khuller et al. [10], in 1996, proved that a graph G with β(G) = 2 can have neither K5 nor K3,3 as its subgraph. In this paper, we obtain a forbidden subgraph...

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008...

In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) a connected graph Γ. Let dΓ(S) be Steiner k-distance S⊆V(Γ). The Wiener k-index or k-center indexSWk(Γ) Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). distanceSDDk(Γ) SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) vertex v in this paper, we consider Nordhaus–Gaddum-type results for SWk(Γ) SDDk(Γ). Upper bounds on SW...

The Laplacian spectra are the eigenvalues of Laplacian matrix L(G) = D(G) - A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical proper...

A k-dominating set of a graph G is a set S of vertices of G such that every vertex outside of S has k neighbors in S. The k-domination number of G, written γk(G), is the size of the smallest k-dominating set in G. In this paper, we derive sharp upper and lower bounds on γk(G) + γk(G) and γk(G)γk(G), where G is the complement of G. We use the results for k = 2 to prove a conjecture of Alon, Balo...

Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)=\{v_{1},v_{2},\cdots, v_{n}\}$ and edge $E(G)$. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ of $G$ is the square order $n$ whose $(i,j)$-entry equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, 0 otherwise, where $d_{i}$ denotes degree $v_{i}$ in $G$. In this paper, we study relationship between index $SO_{p}(G)$ b...