It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for theMerrifield–Simmons indices of u...

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by ...

Let G be a graph with n(G) vertices and m(G) be its matching number. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero of adjacency matrix of G. It is well known that if G is a tree, then η(G) = n(G)− 2m(G). Guo et al. [Jiming GUO, Weigen YAN, Yeongnan YEH. On the nullity and the matching number of unicyclic graphs. Linear Alg. Appl., 2009, 431: 1293–1301] proved tha...

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...

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...

In this paper we will apply the tensor and its traces to investigate spectral characterization of unicyclic graphs. Let $G$ be a graph $G^m$ $m$-th power (hypergraph) $G$. The spectrum is referring adjacency matrix, tensor. called determined by high-ordered spectra (DHS for short) if, whenever $H$ such that $H^m$ cospectral with all $m$, then isomorphic first give formulas graphs, provide some ...