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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 \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} 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.

In this paper the problem of maximizing vertex-degree function index H f ( G ) for trees and unicyclic graphs G of order n and independence number s is solved for strictly convex functions f ( x ) . In the case of unicyclic graphs f ( x ) must also satisfy strict inequality f ( 4 ) + 3 f ( 2 ) > 3 f ( 3 ) + f ( 1 ) . These conditions are fulfilled by general first Zagreb index 0 R α ( G ) for α > 2 , second multiplicative Zagreb index ∏ 2 ( G ) and sum lordeg index S L ( G ) . The extremal graphs are unique and they are stars or have diameter equal to three or to four. The same results are valid for the corresponding minimization problem when f ( x ) is strictly concave and the inequality is reversed.

We study the general Randic index R a ( G ) = ∑ u v ∈ E ( G ) [ d e g G ( u ) d e g G ( v ) ] a , where a ∈ R , E ( G ) is the edge set of a graph G , and d e g G ( u ) and d e g G ( v ) are the degrees of vertices u and v , respectively. For a set of unicyclic graphs of given order and diameter, we present the unique graph having the minimum general Randic index, where − 0 . 64 ≤ a 0 . Since R − 1 2 ( G ) is the Randic index of a graph G , our result holds also for the classical Randic index.

Let G be a connected graph, we define D Q ( G ) = T r ( G ) + D ( G ) as distance signless Laplacian matrix of G, where T r ( G ) and D ( G ) are diagonal matrix with vertex transmissions of G and distance matrix of G, respectively. In this paper, we characterize the extremal graphs which maximize the D Q -spectral radius among complements of unicyclic graphs and trees, respectively. And we also characterize the unique graph among complements of unicyclic graphs of diameter three which maximize the least D Q -eigenvalues.

Given a graph G, the burning number of G is the smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ is a burning sequence of G. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A t-unicyclic graph is a unicycle graph in which the unique vertex in the graph with degree greater than two has degree $$ t + 2 $$ . In this paper, we first present the bounds for the burning number of t-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for $$t\le 2$$ .

The n -vertex unicyclic graphs with extreme values of arithmetic–geometric index or AG index are determined for all values of n . The case regarding the extremal values of the AG index in the class of bicyclic graphs has been stated as open problem.

We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length m ≥ 3 with chains of length s ≥ 1 attached to each vertex of the cycle. There is a one-to-one correspondence between the vertices of the graph and the analyzed subspaces. If an edge connects two vertices, then the angle between the subspaces is equal to 𝜓 𝜖 (0; 𝜋/2); otherwise, the subspaces are orthogonal. Applying the theorem on reduction of unicyclic graph, we prove that nontrivial configurations exist if and only if cos 𝜓 𝜖 (0; 𝜏m,s]. We also deduced formulas for 𝜏m,s and showed that $$ \bigcup_{m,s}\left(\left.0;{\tau}_{m,s}\right]=\right.\left(\left.0;2/5\right]\right. $$ .

We study the generalized k-connectivity κk(G) as introduced by Hager in 1985, as well as the more recently introduced generalized k-edge-connectivity λk(G). We determine the exact value of κk (G) and λk (G) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3.

For α ∈ R , the general sum-connectivity index of a graph G is defined as χ α ( G ) = ∑ u v ∈ E ( G ) [ d e g G ( u ) + d e g G ( v ) ] α , where E ( G ) is the edge set of G and d e g G ( v ) is the degree of a vertex v in G . Let U n , d be the set of unicyclic graphs with n vertices and diameter d .We present the graph with the smallest general sum-connectivity index among the graphs in U n , d for − 1 ≤ α 0 and the graph with the largest general sum-connectivity index among the graphs in U n , d for 0 α 1 . Sharp lower bounds on the classical sum-connectivity index and the harmonic index for graphs in U n , d follow from the lower bound on the general sum-connectivity index.

A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. The question about which unicyclic graphs are integral remains open. We contribute to this problem by presenting three infinite families of integral unicyclic graphs. These families are generated by distinct particular solutions of a Diophantine equation. We also show that two integral unicyclic graphs found through a computer search that do not belong to the families we present are unique with their shapes. Necessary conditions for certain unicyclic graphs to be integral are also given.

Let G be a graph with set of vertices $$V\left( G\right)$$ and set of edges $$E\left( G\right)$$ . The Sombor index is a recently introduced vertex-degree-based-topological index, defined as $$SO(G) = \sum\limits_{uv \in E(G)} {\sqrt {{{({d_u})}^2} + {{({d_v})}^2}} } ,$$ where $$d_{u}$$ denotes the degree of the vertex u. In this paper we study the extremal values of SO over the set of unicyclic graphs and over the set of bicyclic graphs.

"General multiplicative Zagreb indices generalize well-known multiplicative Zagreb indices of graphs. We present lower and upper bounds on the general multiplicative Zagreb indices for unicyclic graphs with given number of vertices and diameter/number of pendant vertices/cycle of given length. All bounds are best possible. Bounds on the classical multiplicative Zagreb indices of unicyclic graphs are corollaries of the general results. "

The max–min rodeg ( $${M\!m_{sde}}$$ ) index is a useful topological index in mathematical chemistry. Damir Vuki $$\check{\text {c}}$$ evi $$\acute{\text {c}}$$ studied the mathematical properties of the max–min rodeg index. In this paper, we determine the n-vertex trees with the second, the third and the fourth for $$n\ge 7$$ , and the fifth for $$n\ge 10$$ minimum $$M\!m_{sde}$$ indices, unicyclic graphs with the second and the third for $$n\ge 5$$ , the fourth, the fifth and the sixth for $$n\ge 7$$ , and the seventh for $$n\ge 9$$ minimum $$M\!m_{sde}$$ indices, and bicyclic graphs with the first for $$n\ge 4$$ , the second and the third for $$n\ge 6$$ , and the fourth for $$n\ge 8$$ minimum $$M\!m_{sde}$$ indices.

A k -cone c -cyclic graph is the join of the complete graph K k and a c -cyclic graph (if k = 0 , we get the usual connected graph). A k -apex tree (resp., k -apex unicyclic graph) is defined as a connected graph G with a k -subset V k ⊆ V ( G ) such that G − V k is a tree (resp., unicylic graph), but G − X is not a tree (resp., unicylic graph) for any X ⊆ V ( G ) with | X | k . In this paper, we extend those extremal results and majorization theorems concerning connected graphs of Liu et al. (2019) to k -cone c -cyclic graphs. We also use a unified method to characterize the extremal maximum and minimum results of many topological indices in the class of k -apex trees and k -apex unicyclic graphs, respectively. The later results extend the main results of Javaid et al. (2019); Liu et al. (2020) and partially answer the open problem of Javaid et al. (2019). Except for the new majorization theorem, some new techniques are also established to deal with the minimum extremal results of this paper.

Recently the exponential Randić index $ {{\rm e}^{\chi}} $ was introduced. The exponential Randić index of a graph $ G $ is defined as the sum of the weights $ {{\rm e}^{{\frac {1}{\sqrt {d \left(u \right) d \left(v \right) }}}}} $ of all edges $ uv $ of $ G $, where $ d(u) $ denotes the degree of a vertex $ u $ in $ G $. In this paper, we give sharp lower and upper bounds on the exponential Randić index of unicyclic graphs.

Nowadays, it is an important task to find extremal values on any molecular descriptor with respect to different graph parameters. In a molecular graph, the vertices represent the atoms and the edges represent the chemical bonds in the terms of graph theory. For one thing, the molecular graphs of some chemical compounds are unicyclic graphs or bicyclic graphs, such as benzene compounds, napthalene, cycloalkane, et al. For another, the symmetric division deg index is proven to be a potentially useful molecular descriptor in quantitative structure-property/activity relationships (QSPR/QSAR) studies recently. Therefore, we present the maximum symmetric division deg indices of unicyclic graphs and bicyclic graphs with given matching number. Furthermore, we identify the corresponding extremal graphs.

We consider a kind of Cayley graphs, including the modified bubble-sort graphs M B n , the Cayley graphs generated by lollipop graphs H n , n − 1 and the other Cayley graphs generated by unicyclic triangle free graphs. We get that if we delete linearly many vertices in Cayley graphs generated by transposition unicyclic triangle free graphs, then the resulting graphs has a large connected component that comprises of nearly all residual vertices.

Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.

We find a basis for the null space of the adjacency matrix of unicyclic graphs. Based on the support of such a basis, we determine a null decomposition of these graphs. As an application, we obtain closed formulas for the independence and matching numbers of unicyclic graphs that solely rely on the support of the graph.

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by U , is considered in this article. This article describes the entries of the inverse of the adjacency matrix o...

Let G be a connected graph with vertex set V ( G ) . The degree Kirchhoff index of G is defined as K f ∗ ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u ) d G ( v ) R G ( u , v ) , where d G ( u ) denotes the degree of the vertex u in G , and R G ( u , v ) denotes the resistance distance between vertices u and v in G . In this paper, we determine maximum degree Kirchhoff index of n -vertex unicyclic graphs with fixed maximum degree, the first seven maximum degree Kirchhoff indices of n -vertex unicyclic graphs, and the corresponding graphs whose degree Kirchhoff indices achieve these values.

The Kirchhoff index K f (G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of K f ( $$\overline{T}$$ ) and the first four maximum values of K f (Ū), where $$\overline{T}$$ and Ū are the complements of a tree T and unicyclic graph U, respectively.

The edge-Szeged index of a graph G is defined as S z e ( G ) = ∑ u v ∈ E ( G ) m u ( u v | G ) m v ( u v | G ) , where m u ( u v | G ) (resp., m v ( u v | G ) ) is the number of edges whose distance to vertex u (resp., v ) is smaller than the distance to vertex v (resp., u ), respectively. In this paper, we characterize the graphs with minimum edge-Szeged index among all the unicyclic graphs with given order and perfect matchings.

The first general multiplicative Zagreb index of a graph G is defined as $$P_1^a (G) = \prod _{v \in V(G)} (deg_G (v))^a$$ and the second general multiplicative Zagreb index is $$P_2^a (G) = \prod _{v \in V(G)} (deg_G (v))^{a \, deg_G (v)}$$ , where V(G) is the vertex set of G, $$deg_{G} (v)$$ is the degree of v in G and $$a \ne 0$$ is a real number. We present lower and upper bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs of given order with a perfect matching. We also obtain lower and upper bounds for trees and unicyclic graphs of given order and matching number. All the trees and unicyclic graphs which achieve the bounds are presented, thus our bounds are sharp. Bounds for the classical multiplicative Zagreb indices are special cases of our theorems and those bounds are new results as well.

We introduce the general eccentric connectivity index of a graph G , E C I α ( G ) = ∑ v ∈ V ( G ) e c c G ( v ) d G α ( v ) for α ∈ R , where V ( G ) is the vertex set of G , e c c G ( v ) is the eccentricity of a vertex v and d G ( v ) is the degree of v in G . We present lower and upper bounds on the general eccentric connectivity index for trees of given order, trees of given order and diameter, and trees of given order and number of pendant vertices. Then we give lower and upper bounds on the general eccentric connectivity index for unicyclic graphs of given order, and unicyclic graphs of given order and girth. The upper bounds for trees of given order and diameter, and trees of given order and number of pendant vertices hold for α > 1 . All the other bounds are valid for 0 α ≤ 1 or 0 α 1 . We present all the extremal graphs, which means that our bounds are best possible.

In this note, we point out some problems in the proofs and the main results of Javaid, Jamil and Tomescu (2019), and we also give the correct versions of them.

Let G be a nontrivial connected graph with a vertex-coloring c: $$V(G)\rightarrow \{1,2,\ldots ,q\},q\in N$$ . For a set $$S\subseteq V(G)$$ and $$|S|\ge 2$$ , a subtree T of G satisfying $$S\subseteq V(T)$$ is said to be an S-Steiner tree or simply S-tree. The S-tree T is called a vertex-rainbow S-tree if the vertices of $$V(T)\setminus S$$ have distinct colors. Let k be a fixed integer with $$2\le k\le |V(G)|$$ , if every k-subset S of V(G) has a vertex-rainbow S-tree, then G is said to be vertex-rainbow k-tree connected. The k-vertex-rainbow index of G, denoted by $$rvx_{k}(G)$$ , is the minimum number of colors that are needed in order to make G vertex-rainbow k-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.

The reciprocal degree resistance distance index of a connected graph G is defined as RDRG=∑u,v⊆VGdGu+dGv/rGu,v, where rGu,v is the resistance distance between vertices u and v in G. Let Un denote the set of unicyclic graphs with n vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in Un and characterize the corresponding extremal graph.

In this paper, we determine the unique graph among complements of unicyclic graphs which maximizes the distance spectral radius. We also characterize the unique graph among complements of unicyclic graphs of diameter three which maximizes the least distance eigenvalues.

Let g ( G ) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k ( k ≥ 8 ) with perfect matchin...

The ABC matrix of a graph G is a square matrix, where the ( i , j ) -entry A B C i j = d i + d j − 2 d i d j if there is an edge between vertices i and j in G, and A B C i j = 0 otherwise. The ABC spectral radius of G, denoted by ν 1 ( G ) , is the largest eigenvalue of the ABC matrix of G. In this paper, we prove that for a unicyclic graph G of order n ≥ 4 , 2 = ν 1 ( C n ) ≤ ν 1 ( G ) ≤ ν 1 ( S n + e ) , with equality if and only if G ≅ C n for the lower bound, and if and only if G ≅ S n + e for the upper bound. This solves a conjecture by Ghorbani et al.

Let T be the group of all complex numbers z with | z | = 1 . A complex unit gain graph, or simply a T -gain graph, is a triple Φ = ( G , T , φ ) consisting of a graph G = ( V , E ) , the circle group T and a gain function φ : E → → T such that φ ( v i v j ) = φ ( v j v i ) − 1 = φ ( v j v i ) ‾ for any adjacent vertices v i and v j . The adjacency matrix A ( Φ ) of the T -gain graph Φ = ( G , T , φ ) of order n is an n × n complex matrix ( a i j ) , where a i j = a j i ‾ = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 if otherwise. The rank of a T -gain graph Φ, denoted by r ( Φ ) , is the rank of the adjacency matrix of Φ. Yu et al. [24] obtained some properties of inertia indexes of a T -gain graph and they characterized the T -gain unicyclic graphs with small positive or negative index. Lu [8] characterized the T -gain bicyclic graphs with rank 2 , 3 or 4. Motivated by above, we determine the rank of a T -gain unicyclic graph and classify the T -gain graphs with rank 2.

Let G ( n , p ) denote the set of unicyclic graphs with n vertices and p ≥ 1 pendent vertices. Hua and Wang (2007) [4] claim to have determined the graph with minimal energy in G ( n , p ) except for the case when n = p + 5 which was left as an open problem. This case was later solved by Huo et al. (2010) [5] . In this paper, we provide a counter example to show that the main result of Hua and Wang (2007) is not true. We also give the correct form of the characterization of unicyclic graphs with given number of pendent vertices and minimal energy. We make use of the well known Coulson's integral formula for the energy of graphs. Our proof is more general and also includes the case n = p + 5 considered by Huo, Ji and Li (2010).

Given two graphs G and H, the k-colored Gallai–Ramsey number $$gr_k(G : H)$$ is defined to be the minimum integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we consider $$gr_k(K_3 : H)$$ , where H is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai–Ramsey numbers for eight of them have been studied step by step in several papers. We determine all the Gallai–Ramsey numbers for the remaining five graphs, and we also obtain some related results for a class of unicyclic graphs. As applications, we find the mixed Ramsey spectra $$S(n; H, K_3)$$ for these graphs by using the Gallai–Ramsey numbers.

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 graph is called unicyclic if the graph contains exactly one cycle. Unicyclic graphs with the fourth extremal Wiener indices are characterized. It is shown that, among all unicyclic graphs with n≥8 vertices, C5Sn−4 and C2u1,u2S3,Sn−4 attain the fourth minimum Wiener index, whereas C3u1,u2P3,Pn−4 attains the fourth maximum Wiener index.

Mili\v{c}evi\'{c} \textit{et al.}, in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, $EM_2(G)$, by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

The trace norm of the digraph is defined as the sum of the singular values of its adjacency matrix. We determine the orientations with, respectively, small and large trace norms among orientations of trees and unicyclic graphs, respectively.

We called graph G non-singular if adjacency matrix A (G) of G is non-singular. A connected graph with n vertices and n-1, n and n+1 edges are called the tree, the unicyclic graph and the bicyclic graph. Respectively, as we all know, each connected bicyclic graph must contain ∞(a,s,b) or θ(p,l,q) as the induced subgraph. In this paper, by using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained.

Let G be a unicyclic graph with edge ideal I ( G ) . For any integer s ≥ 1 , we denote the s-th symbolic power of I ( G ) by I ( G ) ( s ) . It is shown that reg ( I ( G ) ( s ) ) = reg ( I ( G ) s ) , for every s ≥ 1 .

The reduced second Zagreb index $$RM_2$$ of a graph G is defined as $$RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$$, where $$d_G(u)$$ is the degree of the vertex u of graph G. Furtula et al. (Discrete Appl Math 178: 83–88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to $$RM_2$$. In this paper, we obtain an upper bound in terms of order n and size m on $$RM_2$$ of $$K_{r+1}$$-free graphs. Also we prove that among all graphs of order n with chromatic number $$\chi $$, the Turan graph $$T_{n,\,\chi }$$ is the unique graph having the maximum $$RM_2$$. Furthermore, we completely characterize the extremal graphs with respect to $$RM_2$$ among all unicyclic graphs of order n with girth g.

A connected graph G = (V(G), E(G)) is called a quasi-unicyclic graph, if there exists u0 ∈ V(G) such that G − u0 is a unicyclic graph. Denote Q(n, d0) = {G: G is a quasi-unicyclic graph of order n with G − u0 being a unicyclic graph and dG(u0) = d0}. Let A(G) be the adjacency matrix of a graph G, and let λ1(G), λ2(G),…, λn(G) be the eigenvalues in non-increasing order of A(G). The number $$\sum\limits_{i = 1}^n {\lambda _i^k(G)} $$ (k = 0,1, …, n−1) is called the k-th spectral moment of G, denoted by Sk (G). Let S (G) = (S0(G), S1(G),…, Sn−1(G)) be the sequence of spectral moments of G. For two graphs G1, G2, we have G1 ≺SG2 if for some k(k = 1,2,…, n−1), and we have Si(G1) = Si(G2) (i = 0,1, …, k−1) and Sk(G1) < Sk(G2). In this paper, we determine the second to the fourth largest quasi-unicyclic graphs, in an S-order, in the set Q(n, d0), respectively.

Signed graphs are graphs whose edges get signs ±1 and, as for unsigned graphs, they can be studied by means of graph matrices. Here we focus our attention to the largest eigenvalue, also known as the index of the adjacency matrix of signed graphs. Firstly we give some general results on the index variation when the corresponding signed graph is perturbed. Also, we determine signed graphs achieving the minimal or the maximal index in the class of unbalanced unicyclic graphs of order n ≥ 3 .

The non-self-centrality number and the total irregularity of a connected graph G are defined as N ( G ) = ∑ | e G ( v i ) − e G ( v j ) | and i r r t ( G ) = ∑ | d e g G ( v j ) − d e g G ( v i ) | , with summations embracing all pairs of vertices, degG(vi) and ɛG(vi) denoting the degree and eccentricity of the vertex vi. In this paper, we show that there exists a graph G with diameter d such that irrt(G) > N(G) for any integer d ≥ 2. This gives a complete proof of Theorem 10 in Xu et al. (2018), where Xu et al. did not prove it really for d ≥ 4. Moreover, we prove that N(T) > irrt(T) for any tree T of order n ≥ 10 with diameter d ≥ 2 + 26 11 n and maximum degree 4 avoiding degree 3, determine all trees(unicyclic graphs) and with diameter 3 and irrt(T) > N(T) and give a sufficient condition for trees with diameter 4 and irrt(T) > N(T). These partially solve Problems 26 and 27 in the above-mentioned literature.

The first and second Zagreb indices of a graph G are defined as: M 1 ( G ) = ∑ v ∈ V ( G ) d ( v ) 2 and M 2 ( G ) = ∑ u v ∈ E ( G ) d ( u ) d ( v ) , where d ( v ) is the degree of the vertex v . A graph G is said to be a quasi unicyclic graph, if there exists a vertex z ∈ V ( G ) , such that G − z is a unicyclic graph and z is called a quasi vertex. For any integer k ≥ 1 a graph G is called a k -generalized quasi unicyclic graph, if there exists a subset V k ⊆ V ( G ) with cardinality k such that G − V k is a unicyclic graph but for every subset V k − 1 of cardinality k − 1 of V ( G ) , the graph G − V k − 1 is not unicyclic graph. In this paper, we investigate the upper and lower bounds on first and second Zagreb indices for k -generalized quasi unicyclic graphs. Moreover, we characterize the extremal graphs with maximum and minimum Zagreb indices.