Huiqing Liu

Huiqing Liu
Hubei University · Department of Mathematics

PhD

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88
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Publications

Publications (88)
Preprint
For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree $2$. In this paper, we show that if $G$ is a graph of order $n\ge 270$ and $|N(u)\cup N(v)|\geq\frac{n-1}{2}$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$, then $G$ has a HIST, unles...
Preprint
The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote $\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree gr...
Article
An edge colored graph is called rainbow if all the colors on its edges are distinct. A rainbow copy of a graph H in an edge colored graph G is a subgraph of G isomorphic to H such that the coloring restricted to H is rainbow. Let G and H be two graphs. The anti-Ramsey number Ar(G,H) is the maximum number of colors in an edge coloring of G which has...
Article
Let Bn,q be the set of all block graphs with n vertices and all blocks on q+1 vertices for every q≥2. In this paper, we determine the unique graph in Bn,q that attains the minimum spectral radius. This solves a conjecture posted by C.M. Conde, E. Dratman, and L.N. Grippo in 2022.
Article
For H⊆G, the anti-Ramsey number ar(G,H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number ar(Kn,H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was ext...
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An injective k-edge coloring of a graph \(G=(V(G),E(G))\) is a k-edge coloring \(\varphi \) such that if \(e_1\) and \(e_2\) are at distance exactly 2 or in the same triangle, then \(\varphi (e_1)\ne \varphi (e_2)\). The injective chromatic index of G, denoted by \(\chi _i'(G)\), is the minimum k such that G has an injective k-edge coloring. The ed...
Article
The global strong resilience of G with respect to having a fractional perfect matching, also called FSMP number of G, is the minimum number of edges (or resp., edges and/or vertices) whose deletion results in a graph that has no fractional perfect matchings. A graph G is said to be f-fault Hamiltonian if there exists a Hamiltonian cycle in G−F for...
Article
The Funding information section was missing from this article and should have read ‘Partially supported by NSFC under grant number 11971158’.
Preprint
The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the...
Article
Given a graph G, the burning number of G is the smallest integer k for which there are vertices x1,x2,…,xk such that (x1,x2,…,xk) 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...
Article
The expected hitting time Hxy is the expected number of steps it takes for a random walk that starts at x to reach y. For a vertex x of G, the cover cost of x, denoted by CCG(x), is defined as the sum of the expected hitting time for a random walk starting at x to visit all vertices of G. In this paper, we first reveal a close connection between th...
Preprint
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 g...
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Full-text available
Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P|: P i...
Article
Graph burning is a deterministic discrete-time graph process that can be interpreted as a model for the spread of influence in social networks. The burning number of a graph G is the minimum number of steps in a graph burning process for G. It is shown that the graph burning problem is NP-complete even for trees and path forests. In this paper, we...
Article
Connectivity and diagnosability are important parameters in measuring the reliability and fault-tolerance of an interconnection network [Formula: see text]. The [Formula: see text]-extra conditional faulty set [Formula: see text] is a faulty vertex set such that every component of [Formula: see text] has at least [Formula: see text] vertices. The [...
Article
Graph burning is a deterministic discrete time graph process that can be interpreted as a model for the spread of influence in social networks. The burning number b(G) of a graph G is the minimum number of steps in a graph burning process for G. Bonato et al. (2014) conjectured that b(G)≤⌈n⌉ for any connected graph G of order n. In this paper, we c...
Preprint
Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. As a generalization of matching preclusion, the fractional matching preclusion number (FMP number for short) of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional...
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Full-text available
The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional strong matching preclusion number of a graph is the minimum number of edges and/or vertices whose deletion leaves a resulting graph with no fractional perfect matchings. I...
Article
The burning number b(G) of a graph G was introduced by Bonato, Janssen, and Roshanbin [Lecture Notes in Computer Science 8882(2014)] to measure the speed of the spread of contagion in a graph. The graph burning problem is NP-complete even for trees. In this paper, we show that the burning number of any theta graph of order n=q2+r with 1≤r≤2q+1 is e...
Article
A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., \(p(G)=\min \{|{\mathcal {P}}|:\)\({\mathcal {P}}\) is a path cover of...
Article
Fault diagnosis capability is an important metric of the reliability of multiprocessor systems. The h-edge tolerable diagnosability is the maximum number of faulty nodes that the system can guarantee to locate when the number of faulty links does not exceed h. In fact, the 0-edge tolerable diagnosability is exactly the traditional diagnosability. I...
Article
Connectivity and diagnosability are important parameters in measuring the fault-tolerance and reliability of interconnection networks. Given a graph G and a non-negative integer g, the g-extra connectivity of G, denoted by κg(G), is the minimum cardinality of a set of vertices of G, if it exists, whose deletion disconnects G, and every remaining co...
Article
Let G be a connected graph of order n. The signless Laplacian spread of G is defined as \(SQ(G)=q_1(G)-q_n(G)\), where \(q_1(G)\) and \(q_n(G)\) are the maximum and minimum eigenvalues of the signless Laplacian matrix of G, respectively. In this paper, we present some sharp lower bounds for SQ(G) and \(SQ(G)+SQ(G^c)\) in terms of the k-degree and t...
Article
The [Formula presented]-dimensional twisted hypercube [Formula presented] is obtained from two copies of the [Formula presented]-dimensional twisted hypercube [Formula presented] by adding a special perfect matching between the vertices of these two copies of [Formula presented]. The twisted hypercube is a new variant of hypercubes with asymptotica...
Article
Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The g-good-neighbor conditional faulty set is a special faulty set that every fault-free vertex should have at least g fault-free neighbors. The Rg-vertex-connectivity of a connected graph G is the minimum cardinali...
Article
Let G be a graph and \(v\in V(G)\). The neighbors of v, denoted by N(v), are the set of vertices adjacent to v. Let \(N[v]=N(v)\cup \{v\}\) and \(J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}\). A graph G is called quasi-claw-free if \(J(u,v)\ne \emptyset \) for any \(u,v\in V(G)\) with \(d(u,v)=2\). In this paper, we show that if G is...
Preprint
Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The $R^g$-vertex-connectivity of a connected graph $G$ is the minimum cardinality of a faulty set $X\subseteq V(G)$ such that $G-X$ is disconnected and every fault-free vertex has at least $g$ fault-free neighbors....
Article
Let $G$ be a graph and $T$ a certain connected subgraph of $G$. The $T$-structure connectivity $\kappa(G; T)$ (or resp., $T$-substructure connectivity $\kappa^{s}(G; T)$) of $G$ is the minimum number of a set of subgraphs $\mathcal{F}=\{T_{1}, T_{2}, \ldots, T_{m}\}$ (or resp., $\mathcal{F}=\{T^{'}_{1}, T^{'}_{2}, \ldots, T^{'}_{m}\}$) such that $T...
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In this paper, we give some bounds on the signless Laplacian index of graphs in terms of independence number. In addition, these results disprove a conjecture in [3] involving the signless Laplacian index and independence number of graphs.
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Full-text available
Given a graph $G$, denote by $\Delta$ and $\chi^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maxim...
Preprint
Given a graph $G$, denote by $\Delta$ and $\chi^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maxim...
Article
Full-text available
Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is e...
Preprint
Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is e...
Article
A bag Bag(p,q), is a graph obtained from a complete graph K-p by replacing an edge uv by a path P-q. In this paper, we show that for all the connected graphs of order n >= 5 with signless Laplacian index q(1)(G) and radius rad(G), q(1)(G) center dot rad(G) is maximum for and only for the graph Bagn-2s+3,2s-1, where s = [n/4 ]. This solves a conject...
Article
The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value of Kirchhoff index among the unicyclic graphs with fixed number of vertices and maximum degree, and characteri...
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In this paper, we investigate the least eigenvalue of signless Laplacian matrix of a connected graph, and determine the graphs which have the minimum least signless Laplacian eigenvalue among all nonbipartite graphs with given stability number or covering number , respectively.
Article
A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a s...
Article
An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processor and communication links, respectively. Connectivity is an important measurement for the fault tolerant in interconnection network. Two vertices is maximally local-connected if the maximum number of internally vertex-disjoint paths between t...
Article
A bug Bugp,q1,q2Bugp,q1,q2 is a graph obtained from a complete graph KpKp by deleting an edge uv and attaching paths Pq1Pq1 and Pq2Pq2 at u and v , respectively. In this paper, we show that for connected graphs G of order n with signless Laplacian index q1(G)q1(G) and diameter diam(G)diam(G), q1(G)⋅diam(G)q1(G)⋅diam(G) is maximized for and only for...
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In this paper, we consider the edge fault tolerance of nn-dimensional burnt pancake graph BPnBPn such that each vertex is incident with at least two fault free edges. Based on this requirement, we show that BPnBPn contains a fault free Hamilton cycle even it has up to 2n−52n−5 link faults, where n≥3n≥3.
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Full-text available
Let G be a connected graph with n vertices and m edges. Let q1, q2,..., qn be the eigenvalues of the signless Laplacian matrix of G, where q1 ≥ q2 ≥ • • • ≥ qn. The signless Laplacian Estrada index of G is defined as SLEE(G) = Σni=1 eqi. In this paper, we present some sharp lower bounds for SLEE(G) in terms of the k-degree and the first Zagreb inde...
Article
In this paper, we first show that if the second smallest Laplacian eigenvalue of a graph is no less than (k-1)n (δ+1)(n-1-δ) or the second largest signless Laplacian eigenvalue of a graph is no more than 2δ-(k-1)n (δ+1)(n-1-δ), then the graph is k-edge-connected, where δ is the minimum degree of the graph and n is the order of the graph. Also, we g...
Article
The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings, and the conditional matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph with no isolated vertices that has neither perfec...
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In the paper, we will present two sufficient conditions for a bipartite graph to be Hamiltonian and a graph to be traceable, respectively.
Article
A connected graph G=(V,E) is called a quasi-tree, if there exists u0∈V(G) such that G-u0 is a tree. Denote Q(n,d0)={G:G is a quasi-tree graph of order n with G-u0 being a tree 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 ∑i=1nλik(G)(k=0,1,…,...
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In this paper, we present a unified and simple approach to extremal acyclic graphs without perfect matching for the energy, the Merrifield-Simmons index and Hosoya index.
Article
Let Delta >= 3. Denote by T-n.Delta the set of all trees with n vertices and maximum degree Delta and by T*(n,Delta) the set of all Delta-trees with n vertices. In this work, we first show that all Delta-trees come before all trees in T-n.Delta/T*(n.Delta), in an S-order and present a criterion for a Delta-tree coming before another Delta-tree in a...
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Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.
Article
Let G be a simple connected graph and α be a given real number. The zeroth-order general Randić index 0Rα(G) is defined as ∑v∈V(G)[dG(v)]α, where dG(v) denotes the degree of the vertex v of G. In this work, we give, for any α(≠0,1), some sharp bounds on the zeroth-order general Randić index 0Rα of all unicyclic graphs with n vertices and diameter d...
Article
The Wiener index of a graph G is defined as W(G)=∑ u,v d G (u,v), where d G (u,v) is the distance between u and v in G, and the sum goes over all pairs of vertices. In this paper, we characterize the connected unicyclic graph with minimum Wiener indices among all connected unicyclic graphs of order n and girth g with k pendent vertices.
Article
Let G = (V,E) be a graph on n vertices, and let λ 1 ≥ λ 2 ≥ ··· ≥ λ n be eigenvalues of G. The Hückel energy of G, HE(G), is defined as {equation presented} In this paper, we present some new upper bounds for HE(G), from which we can improve some known results.
Article
Let Tn,d be the class of trees with n vertices and diameter d. In this paper, the lexicographic ordering of trees in the set Tn,d(3⩽d⩽n-2) by spectral moments is considered, and the last d2+1 trees, in an S-order, among all trees in Tn,d(4⩽d⩽n-3) are characterized. Moreover, all trees in Tn,d have an S-order for d⩽3 and d⩾n-2.
Article
The Wiener index of a graph G is defined as W(G)=∑ u,v d G (u,v), where d G (u,v) is the distance between u and v in G and the sum goes over all the pairs of vertices. In this paper, we first present the 6 graphs with the first to the sixth smallest Wiener index among all graphs with n vertices and k cut edges and containing a complete subgraph...
Article
Let G0 and G1 be two graphs with the same vertices. The new graph G(G0, G1; M) is a graph with the vertex set V (G0)∪V(G1) and the edge set E(G0)∪E(G1)∪M, where M is an arbitrary perfect matching between the vertices of G0 and G1, i.e., a set of cross edges with one endvertex in G0 and the other endvertex in G1. In this paper, we will show that if...
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Let G be a simple graph with n vertices, m edges, diameter D and degree sequence d 1, d 2, …, d n , and let λ1(G) be the largest Laplacian eigenvalue of G. Denote Δ = max{d i : 1 ≤ i ≤ n}, and , where α is a real number. In this article, we first give an upper bound on λ1(G) for a non-regular graph involving Δ and D; next present two upper bounds o...
Article
Let G be a simple graph with n vertices, m edges. Let Δ and δ be the maximum and minimum degree of G, respectively. If each edge of G belongs to t triangles (t≥1), then we present a new upper bound for the Laplacian spectral radius of G as follows: λ1(G)≤2Δ−t+(2Δ−t)2+8m−4δ(n−1)−4δ2+4(δ−1)Δ2. Moreover, we give an example to illustrate that our resul...
Article
The Randic index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))(-1/2) over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randic index of a triangle-free graph G with given m...
Article
A connected graph G=(V,E) is called a quasi-tree, if there exists u0∈V(G) such that G-u0 is a tree. Denote Q(n,d0)={G:Gis a quasi-tree graph of ordernwithG-u0being a tree anddG(u0)=d0}. In this paper, we determined the maximal and the second maximal spectral radii of all quasi-tree graphs in the set Q(n,d0).
Article
The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we solve a conjecture in Ou, J. Math. Chem, DOI: 10.1007/S10910-006-9199-1 concerning acyclic molecular graphs with maximal Hosoya index and diameter 4.
Article
Let G = (V, E) be a simple graph of order n with V(G) = {ν1, ν2,..., μn} and degree sequence d1, d2,...,dn. Let ρ{G) be the largest eigenvalue of adjacency matrix of G, and let E(G) be the energy of G. Denote ( αt)i = Σi∼j djα and (αm)i = (αt) i/diα, where α is a real number. In this paper, we obtain two sharp bounds on ρ(G) in terms of ( αm)i or (...
Article
The Wiener index of a (molecular) graph is defined as W(G) = Σuv dG(u, v), where dG(u,v) is the distance between u and v in G and the sum goes over all the pairs of vertices. In this paper, we obtain the trees with minimum and second-minimum Wiener indices among all the trees with n vertices and diameter d, respectively.
Article
Given a molecular graph G, the Hosoya index z(G) and the Merrifield-Simmons index σ(G) are defined as the total number of the independent edge-sets and the total number of the independent vertexsets of the graph G, respectively. Let un,g denote the set of unicyclic graphs with order n and girth g > 3. In this paper, we will give the first [g/2] + 1...
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The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))-1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. In this paper, we present a upper bound on the Randić index for all chemical graphs with n vertices, m ≥ n edges and k > 0 pendant vertices, and determine...
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The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we characterize the unicyclic graphs with maximal Merrifield-Simmons indices and minimal Hosoya indices, respectively, among all unic...
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The second Zagreb index M 2(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, Un-3nU_{n-3}^n and C n have the maximum...
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In this paper, we present lower and upper bounds for the independence number α(G) and the clique number ω(G) involving the Laplacian eigenvalues of the graph G.
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For a graph G, let diff(G) = p(G) − c(G), where p(G) and c(G) denote the orders of a longest path and a longest cycle in G, respectively. Let G be a 3-connected graph of order n. In the paper, we give a best-possible lower bound to σ 4(G) to assure diff(G) ≤ 1. The result settles a conjecture in J. Graph Theory 37 (2001), 137–156.
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Let G=(V,E) be a graph with n vertices and e edges. Denote V(G)={v 1,v 2,...,v n }. The 2-degree of v i , denoted by t i , is the sum of degrees of the vertices adjacent to vi, 1\leqslant i\leqslant nv_i, 1\leqslant i\leqslant n. Let σ i be the sum of the 2-degree of vertices adjacent to v i . In this paper, we present two sharp upper bounds f...
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In the paper, we will determine graphs with the maximal spectral radius among all the unicyclic graphs with n vertices and diameter d.
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The Merrifield-Simmons index σ = σ(G) and the Hosoya index z = z (G) of a (molecular) graph G are defined as the total number of the independent vertexsets and the total number of the independent edgesets of the graph G, respectively. Let Jn,d denote the set of trees on n vertices and diameter d. Li, Zhao and Gutman [MATCH Commun. Math. Comput. Che...
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The general Randić index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))α over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G and α is a real number with α ≠ 0. In this paper, we characterize the trees with minimal and maximal general Randić indices, respectively, among all t...
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The Randić index R(G) of a graph G is the sum of the weights (d(u)d(v))-\frac12(d(u)d(v))^{-\frac{1}{2}} of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on th...
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Let G be a simple connected graph with V(G)={v1,v2,…,vn} and girth at least 5. Let Δ be the maximum degree of G. In this paper, we present a new sharp upper bound for the spectral radius of G as follows:ρ(G)⩽-1+4n+4Δ-32.Moreover, equality holds if and only if G≅C5.
Article
The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v))α of all edges uv of G, where α is a real number (α≠0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T). Also, for -1⩽α<0, we give a sharp lower bound and a...
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In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u),d(v)}:d(u,v)=2}=μ, then for each edge e, G has a cycle containing e of length at least min{n,2μ}, unless G is a spanning subgraph of K μ + K c n−μ or K 3+(lK μ −2∪K s ), where n=l(μ−2)+s+3,1≤s≤μ−2.
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The Randi index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))-1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Discrete Mathematics 257, 29–38 by Delorme et al. gave a best-possible lower bound on the Randi index of a triangle-free graph G with given minimum degr...
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A cycle C of a graph G is dominating if each component of GnC is edgeless. In the paper, we will give two sufficient conditions for each longest cycle of a 3-connected graph to be a dominating cycle.
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Let G be a connected graph of order n. A dominating set in G is a subset S of V(G) such that each element of V(G) − S is adjacent to a vertex of S. The least cardinality of a dominating set is the domination number. In the paper, we will give bounds of the Laplacian spectrum of G involving the domination number.
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The general Randić index wα(G) of a graph G is the sum of the weights (d(u)d(v))α of all edges uv of G, where α is a real number and d(u) denotes the degree of the vertex u. Let F n,m be the set of all trees on n vertices with a maximum matching of cardinality m. Denote by Tn,m0 the tree on n vertices obtained from the star graph Sn,m+1 by attachin...
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We study the spectral radius of graphs with n vertices and k cut edges. In this paper, we show that of all the connected graphs with n vertices and k cut edges, the maximal spectral radius is obtained uniquely at Knk, where Knk is a graph obtained by joining k independent vertices to one vertex of Kn−k. We also discuss the limit point of the maxima...
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A graph G is said to be cyclable if for each orientation (G) over right arrow of G, there exists a set S of vertices such that reversing all the arcs of (G) over right arrow with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n greater than or equal to 36. In this paper, we show that if for any three independen...
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Let G be a simple graph with n vertices and m edges and Gc be its complement. Let δ(G)=δ and Δ(G)=Δ be the minimum degree and the maximum degree of vertices of G, respectively. In this paper, we present a sharp upper bound for the Laplacian spectral radius as follows:λ1(G)⩽(Δ+δ−1)+(Δ+δ−1)2+4(4m−2δ(n−1))2.Equality holds if and only if G is a connect...

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