The ordering of unicyclic graphs with the smallest algebraic connectivity

Fielder [M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305] has turned out that G is connected if and only if its algebraic connectivity a(G)>0. In 1998, Fallat and Kirkland [S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74] posed a conjecture: if G is a connected graph on n vertices with girth g≥3, then a(G)≥a(C_n,g) and that equality holds if and only if G is isomorphic to C_n,g. In 2007, Guo [J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with vertices.     

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.     

Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U(n,d) be the set of unicyclic graphs on n vertices with diameter d. In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U(n,d). It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449-457].     

Tricyclic graphs with maximum Merrifield-Simmons index

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 the Merrifield-Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246-256]. The sharp upper bound for the Merrifield-Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield-Simmons index in (n,n+1)-graphs, J. Math. Chem. 43 (1) (2008) 75-91]. The sharp lower bound for the Merrifield-Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield-Simmons index is determined.     

The Laplacian spread of quasi-tree graphs

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. Bao, Tan and Fan [Y.H. Bao, Y.Y. Tan,Y.Z. Fan, The Laplacian spread of unicyclic graphs, Appl. Math. Lett. 22 (2009) 1011-1015.] characterize the unique unicyclic graph with maximum Laplacian spread among all connected unicyclic graphs of fixed order. In this paper, we characterize the unique quasi-tree graph with maximum Laplacian spread among all quasi-tree graphs in the set Q(n,d) with .     

The number of independent sets in unicyclic graphs with a given diameter

Let U_n,d denote the set of unicyclic graphs with a given diameter d. In this paper, the unique unicyclic graph in U_n,d with the maximum number of independent sets, is characterized.     

On the nullity of graphs with pendent vertices

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60-67] characterized the extremal graphs attaining the upper bound n-2 and the second upper bound n-3. In this paper, as the continuance of it, we determine the extremal graphs with pendent vertices achieving the third upper bound n-4 and fourth upper bound n-5. We then proceed recursively to construct all graphs with pendent vertices which satisfy η(G)>0. Our results provide a unified approach to determine n-vertex unicyclic (respectively, bicyclic and tricyclic) graphs which achieve the maximal and second maximal nullity and characterize n-vertex extremal trees attaining the second and third maximal nullity. As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.     

On the nullity and the matching number of unicyclic graphs

Let G be a graph with n vertices and ν(G) be the matching number of G. Let η(G) denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G)=n-2ν(G). Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212-220] proved that the nullity set of all unicyclic graphs with n vertices is {0,1,…,n-4} and characterized the unicyclic graphs with η(G)=n-4. In this paper, we characterize the unicyclic graphs with η(G)=n-5, and we prove that if G is a unicyclic graph, then η(G) equals , or n-2ν(G)+2. We also give a characterization of these three types of graphs. Furthermore, we determine the unicyclic graphs G with η(G)=0, which answers affirmatively an open problem by Tan and Liu.     

Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

In this paper, we consider the following problem. Over the class of all simple connected graphs of order n with k pendant vertices (n, k being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.     

The smallest values of algebraic connectivity for trees

The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let ■n be the set of all trees of order n. In this paper, we will provide the ordering of trees in ■n up to the last eight trees according to their smallest algebraic connectivities when n ≥ 13. This extends the result of Shao et al. [The ordering of trees and connected graphs by algebraic connectivity. Linear Algebra Appl., 428, 1421-1438 (2008)].     

The Algebraic Connectivity of Graphs with Given Matching Number

The algebraic connectivity of a graph is the second smallest eigenvalue of the associated Laplacian matrix. In this paper, we not only characterize the extremal graphs with the maximal algebraic connectivity among all graphs of order n with given matching number, but also determine the extremal tree with the maximal algebraic connectivity among all trees of order n with given matching number.     

LARGEST EIGENVALUE OF A UNICYCLIC MIXED GRAPH

The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ1 (U)=n or λ1 (U)∈ (n ,n 1] are characterized.     

Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph K_m,n,r are developed. Using these, we obtain the chromatic equivalence classes for K_m,n,n (where 1≤m≤n) and K_m1,m2,m3 (where |m_i−m_j|≤3). In particular, it is shown that (i) K_m,n,n (where 2≤m≤n) and (ii) K_m1,m2,m3 (where |m_i−m_j|≤3, 2≤m_i,i=1,2,3) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175-179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27-37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs K_n1,n2,n3 J. Systems Sci. Math. Sci. 20 (2000) 181-186].     

The convex hull of degree sequences of signed graphs

As shown in [D. Hoffman, H. Jordon, Signed graph factors and degree sequences, J. Graph Theory 52 (2006) 27-36], the degree sequences of signed graphs can be characterized by a system of linear inequalities. The set of all n-tuples satisfying this system of linear inequalities is a polytope P_n. In this paper, we show that P_n is the convex hull of the set of degree sequences of signed graphs of order n. We also determine many properties of P_n, including a characterization of its vertices. The convex hull of imbalance sequences of digraphs is also investigated using the characterization given in [D. Mubayi, T.G. Will, D.B. West, Realizing degree imbalances of directed graphs, Discrete Math. 239 (2001) 147-153].     

On the existence of edge cuts leaving several large components

We characterize graphs of large enough order or large enough minimum degree which contain edge cuts whose deletion results in a graph with a specified number of large components. This generalizes and extends recent results due to Ou [Jianping Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005), 365-371] and Zhang and Yuan [Z. Zhang, J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005), 128-134].     

Caterpillar arboricity of planar graphs

We solve a conjecture of Roditty, Shoham and Yuster [P.J. Cameron (Ed.), Problems from the 17th British Combinatorial Conference, Discrete Math., 231 (2001) 469-478; Y. Roditty, B. Shoham, R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411-417] on the caterpillar arboricity of planar graphs. We prove that for every planar graph G=(V,E), the edge set E can be partitioned into four subsets (E_i)_1?i?4 in such a way that G[E_i], for 1?i?4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.     

Matroids with few non-common bases

In [On Mills's conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271-276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195-205]: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂≤k. In [Matroids with many common bases, Discrete Math. 270 (2003) 193-205], Lemos proved a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected. In this paper, we extend these results.     

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n,?d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n,?d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].     

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186-209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253-267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown.We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time.Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.     

Further analysis of the number of spanning trees in circulant graphs

Let 1?s₁<s₂<?<s_k?⌊n/2⌋ be given integers. An undirected even-valent circulant graph, has n vertices 0,1,2,…, n-1, and for each and j(0?j?n-1) there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]], is that where a_n satisfies a linear recurrence relation of order 2^s_k-1. And, most recently, for odd-valent circulant graphs, a nice investigation on the number a_n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69-79].In this paper, we explore further properties of the numbers a_n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a_n, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.