Let η(G) be the number of connected induced subgraphs in a graph G, and Ḡ complement G. We prove that + η(Ḡ) is minimum, among all n-vertex graphs, if only G has no path on four vertices. Since star Sn with maximum degree n − 1 unique tree diameter 2, minimum trees, while shown to achieved by whose sequence (⌈n/2⌉, ⌊n/2⌋, 1, . , 1). Furthermore, we every order ≥ 5 must have at most 3, cut verte...

the concept of geometric-arithmetic indices (ga) was put forward in chemical graph theoryvery recently. in spite of this, several works have already appeared dealing with these indices.in this paper we present lower and upper bounds on the second geometric-arithmetic index(ga2) and characterize the extremal graphs. moreover, we establish nordhaus-gaddum-typeresults for ga2.

Let D=(V(D),A(D)) be a finite, simple digraph and k positive integer. A function f:V(D)→{0,1,2,…,k+1} is called [k]-Roman dominating (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} AN−[v]=AN−(v)∪{v}. The weight of [k]-RDF f ω(f)=∑v∈V(D)f(v). minimum on D the domination number, denoted by γ[kR](D). For k=2 k=3, we call them double Roman number tr...

An edge-colored graph G is conflict-free connected if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the smallest number of colors that are needed in order to make G conflict-free connected. In this paper, we determine all trees T of ord...

A Grundy coloring of a graph G is a proper vertex coloring of G where any vertex x, colored with c(x), has a neighbor of any color 1, 2, . . . , c(x)− 1. A central graph Gc is obtained from G by adding an edge between any two non adjacent vertices in G and subdividing any edge of G once. In this note we focus on Grundy colorings of central graphs. We present some bounds related to parameters of...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

For a simple graph G with chromatic number x(G), the Nordhaus-Gaddum inequalities give upper and lower bounds for z(G)•(G ¢) and z(G)+ x(GC). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A(G)B(G c) is minimum, where A and B are each of chromatic number, achromatic number and pseud...

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extrem...

For a graph G with vertex set V , the total redundance, TR(G), and efficiency, F (G), are defined by the two expressions: TR(G) = min{ ∑ v∈S (1+deg v) : S ⊆ V and |N [x]∩S| ≥ 1 ∀x ∈ V }, F (G) = max{ ∑ v∈S (1 + deg v) : S ⊆ V and |N [x]∩ S| ≤ 1 ∀x ∈ V }. That is, TR measures the minimum possible amount of domination if every vertex is dominated at least once, and F measures the maximum number o...