For a graph G, the Merrifield-Simmons index i(G) is defined as the total number of independent sets of the graph G. Let G(n, l, k) be the class of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. In this paper, we characterize the unique unicyclic graph possessing prescribed girth and pendent vertices with the maximal Merrifield-Simmons index among all graphs ...

The harmonic index of a graph G is defined as the sum of weights 2 d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, we determine the graph with minimum harmonic index among all unicyclic graphs with a perfect matching. Moreover, the graph with minimum harmonic index among all unicyclic graphs with a given matching num...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

In this paper, we will describe some algorithms and give their complexity as following: (1) The algorithm for finding a dominating set of radius r in a vertex-weighted graph with small number of spanning tress. The complexity of this algorithm for the unicyclic graph is O(m.n). (2) The algorithm for finding an absolute and vertex p-center of a vertex-weighted graph with small number of spanning...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...

A set S ⊆ V is independent in a graph G = (V,E) if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. A connected graph having only one cycle, say C, is a unicyclic graph. In this paper we prove that if G is a uni...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, th...

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let Un,d be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from Un,d minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hos...