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Mostar indices are recently proposed distance-based graph invariants, that already have been much investigated and found applications. In this paper, we investigate the inverse problem for Mostar indices of unicyclic and bicyclic molecular graphs. We prove that all positive integers other than 1, 2, 3, and 5 can be the Mostar index of some bicyclic molecular graph. In addition, we resolve the inverse edge Mostar index problem for molecular unicyclic and bicyclic graphs, and in doing so, establish the second and third smallest value of the edge Mostar index of unicyclic graphs.

Let $G$ be a connected graph and $W=\{w_1, w_2,\ldots,w_k\}$ be an ordered subset of vertices of $G$. For any vertex $v$ of $G$, the ordered $k$-vector $$r(v|W)=(d(v,w_1), d(v,w_2),\ldots,d(v,w_k))$$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct metric representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension denoted by $\dim(G)$. A resolving set $W$ is called a non-isolated resolving set for $G$ if the induced subgraph $\langle W\rangle$ of $G$ has no isolated vertices. The minimum cardinality of a non-isolated resolving set for $G$ is called the non-isolated resolving number of $G$ and denoted by $nr(G)$. The aim of this paper is to find properties of unicyclic graphs that have non-isolated resolving number $2$ and then to characterize all these graphs.

AbstractFor a connected simple graph G, the inverse degree index and forgotten index are defined as [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the degree of vertex u in G. In this article, we use some graph transformations and introduce a method for determining the extremal values of inverse degree index and forgotten index on the class of unicyclic graphs.

The graph in this paper is a simple and connected graph with V(G) is vertex set and E(G) is edge set. An inklusif local irregularity vertex coloring is defined should be maping l:V(G) í {1,2,..., k} as vertex labeling and wi : V(G) í N is function of inclusive local irregularity vertex coloring, with wi(v) = l(v) + ∑u∈N(v) l(u) in other words, an inclusive local irregularity vertex coloring is to assign a color to the graph with the resulting weight value by adding up the labels of the vertices that are should be neighboring to its own label. The minimum number of colors produced from inclusive local irregularity vertex coloring of graph G is called inclusive chromatic number local irregularity, denoted by Xlisi(G). Should be in this paper, we learn about the inclusive local irregularity vertex coloring and determine the chromatic number on unicyclic graphs.

The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as $ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $ Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.

A graph G admits a graceful labeling if there is a one-to-one map f from the set of vertices of G to such that when an edge xy is assigned the label the resulting set of edge labels is When such a labeling exists, G is called graceful. Rosa showed that a cycle Cn () is graceful if and only if n is congruent to 0 or 3 modulo 4. Truszczyński conjectured that unicyclic graphs, except the cycles in Rosa’s result are graceful. Several classes of unicyclic graphs are known to be graceful. In this article, we present a survey of results related to Truszczyński’s conjecture. We also present a new class of graceful unicyclic graphs.

A tadpole (also a canoe paddle or lollipop) is a graph that arises from a cycle and a path by gluing a terminal vertex of the path to an arbitrary vertex of the cycle. In this article, we show that all tadpoles factorize the complete graph if n is odd. We use methods similar to those used for isomorphic factorizations of complete graphs into spanning trees. In of this article, we show that our methods do not work for isomorphic factorizations of into tadpoles if n is even.

For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees of the vertices. This paper shows a simple and unified approach to the greatest SK indices for unicyclic graphs by using some transformations and characterizes these graphs with the first, second, and third SK indices having order r ≥ 5 and girth g ≥ 3, where girth is the length of the shortest cycle in a graph.

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph Γ, the Lanzhou index (Lz index) is obtained by the sum of deg(v)2de¯g(v) over all the vertices, where deg(v) and de¯g(v) are degrees of the vertex v in Γ and its complement Γ¯ respectively. Let Vαβ be a class of unicyclic graphs (same order and size) such that each graph of this class has order α and β leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs Vαβ. Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs.

Given a simple connected graph G, the metric dimension dim(G) (and edge metric dimension edim(G)) is defined as the cardinality of a smallest vertex subset S⊆V(G) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S. It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim(G) and edim(G) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈E(u) f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G,x)=∑k=0ns(G,k)xk$I(G,x) = \sum\nolimits_{k = 0}^n {s\left({G,k} \right)x^k }$, where s(G, k) is the number of independent sets of G with size k and s(G, 0) = 1. A unicyclic graph is a graph containing exactly one cycle. Let Cn be the cycle on n vertices. In this paper we study the independence polynomial of unicyclic graphs. We show that among all connected unicyclic graphs G on n vertices (except two of them), I(G, t) > I(Cn, t) for sufficiently large t. Finally for every n ≥ 3 we find all connected graphs H such that I(H, x) = I(Cn, x).

The harmonic index of a graph G is defined as the sum of the weights 2d(u)+d(v)${2 \over {d(u) + d(v)}}$ of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic index for unicyclic graphs with given diameter and characterize the corresponding extremal graphs. This answers an unsolved problem of Zhu and Chang [26].

The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.

Abstract The eigenvalues of G are denoted by λ 1 ( G ) , λ 2 ( G ) , … , λ n ( G ) $\lambda_{1}(G), \lambda_{2}(G), \ldots, \lambda_{n}(G)$ , where n is the order of G. In particular, λ 1 ( G ) $\lambda _{1}(G)$ is called the spectral radius of G, λ n ( G ) $\lambda_{n}(G)$ is the least eigenvalue of G, and the spread of G is defined to be the difference between λ 1 ( G ) $\lambda_{1}(G)$ and λ n ( G ) $\lambda_{n}(G)$ . Let U ( n ) $\mathbb{U}(n)$ be the set of n-vertex unicyclic graphs, each of whose vertices on the unique cycle is of degree at least three. We characterize the graphs with the kth maximum spectral radius among graphs in U ( n ) $\mathbb{U}(n)$ for k = 1 $k=1$ if n ≥ 6 $n\ge6$ , k = 2 $k=2$ if n ≥ 8 $n\ge8$ , and k = 3 , 4 , 5 $k=3,4,5$ if n ≥ 10 $n\ge10$ , and the graph with minimum least eigenvalue (maximum spread, respectively) among graphs in U ( n ) $\mathbb{U}(n)$ for n ≥ 6 $n\ge6$ .

In this paper, we investigate the new graph characteristic, the edge irregularity strength, denoted as es, as a modification of the well known irregularity strength, total edge irregularity strength and total vertex irregularity strength. As a result, we obtain the exact value of an edge irregularity strength of corona product of cycle with isolated vertices.

A vertex w of a connected graph G strongly resolves two distinct vertices u,v∈V(G), if there is a shortest u,w path containing v, or a shortest v,w path containing u. A set S of vertices of G is a strong resolving set for G if every two distinct vertices of G are strongly resolved by a vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. To study the strong metric dimension of graphs, a very important role is played by a structure of graphs called the strong resolving graph In this work, we obtain the strong metric dimension of some families of cactus graphs, and along the way, we give several structural properties of the strong resolving graphs of the studied families of cactus graphs.

For any simple graph G, let D(G) denote the degree set {degG(v) : v ∈ V (G)}. Let S be a finite, nonempty set of positive integers. In this paper, we first determine the families of graphs G which are unicyclic, bipartite satisfying D(G) = S, and further obtain the graphs of minimum orders in such families. More general, for a given pair (S, T) of finite, nonempty sets of positive integers of the same cardinality, it is shown that there exists a bipartite graph B(X, Y ) such that D(X) = S, D(Y ) = T and the minimum orders of different types are obtained for such graphs

The Hosoya index of a graph is defined by the total number of the matchings of the graph. In this paper, we determine the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. Our results somewhat answer a question proposed by Wagner and Gutman in 2010 for unicyclic graphs with small diameter.

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.

The first ($Pi_1$) and the second $(Pi_2$) multiplicative Zagreb indices of a connected graph $G$, with vertex set $V(G)$ and edge set $E(G)$, are defined as $Pi_1(G) = prod_{u in V(G)} {d_u}^2$ and $Pi_2(G) = prod_{uv in E(G)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. In this paper we present a simple approach to order these indices for connected graphs on the same number of vertices. Moreover, as an application of this simple approach, we extend the known ordering of the first and the second multiplicative Zagreb indices for some classes of connected graphs.

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 considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.