solution to a conjecture on the maximal energy of bipartite bicyclic graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let c n denote the cycle of order n and p n 6 6 the graph obtained from joining two cycles c 6 by a path p n 12 with its two leaves let b n denote the class of all bipartite bicyclic graphs but not the graph r a b which is obtained from joining two cycles c a and c b a b 10 and a b 2 mod 4 by an edge in i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005 gutman and vidovic conjectured that the bicyclic graph with maximal energy is p n 6 6 for n 14 and n 16 in x li j zhang on bicyclic graphs with maximal energy linear algebra appl 427 2007 87 98 li and zhang showed that the conjecture is true for graphs in the class b n however they could not determine which of the two graphs r a b and p n 6 6 has the maximal value of energy in b furtula s radenkovic i gutman bicyclic molecular graphs with the greatest energy j serb chem soc 73 4 2008 431 433 numerical computations up to a b 50 were reported supporting the conjecture so it is still necessary to have a mathematical proof to this conjecture this paper is to show that the energy of p n 6 6 is larger than that of r a b which proves the conjecture for bipartite bicyclic graphs for non bipartite bicyclic graphs the conjecture is still open

the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

unicyclic graphs with large energy we study the energy i e the sum of the absolute values of all eigenvalues of so called tadpole graphs which are obtained by joining a vertex of a cycle to one of the ends of a path by means of the coulson integral formula and careful estimation of the resulting integrals we prove two conjectures on the largest and second largest energy of a unicyclic graph due to caporossi cvetkovic gutman and hansen and gutman furtula and hua respectively moreover we characterise the non bipartite unicyclic graphs whose energy is largest

finding low rank solutions of sparse linear matrix inequalities using convex optimization this paper is concerned with the problem of finding a low rank solution of an arbitrary sparse linear matrix inequality lmi to this end we map the sparsity of the lmi problem into a graph we develop a mathematical framework to relate the rank of the minimum rank solution of the lmi problem to the sparsity of its underlying graph furthermore we propose three graph theoretic convex programs to obtain a low rank solution two of these convex optimization problems are based on a tree decomposition of the sparsity graph the third one does not rely on any computationally expensive graph analysis and is always polynomial time solvable at the cost of offering a milder theoretical guarantee on the rank of the obtained solution compared to the other two methods the results of this work can be readily applied to three separate problems of minimum rank matrix completion conic relaxation for polynomial optimization and affine rank minimization the results are finally illustrated on two applications of opt

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

unicyclic graphs with maximal energy abstract let 1 2 n be the eigenvalues of a graph g of order n the energy of g is defined as e g 1 2 n let p n 6 6 be the graph obtained from two copies of c 6 joined by a path p n 10 b n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles c a and c b a b 10 and a b 2 mod 4 joined by an edge in this paper we show that p n 6 6 is the graph with maximal energy in b n which gives a partial solution to gutman s conjecture in gutman and vidovic 2001 i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005

on the graph complement conjecture for minimum semidefinite rank in this note we combine a number of recent ideas to give new results on the graph complement conjecture for minimum semidefinite rank

on the maximal energy tree with two maximum degree vertices abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix for 3 and t 3 denote by t a t or simply t a the tree formed from a path p t on t vertices by attaching 1 p 2 s on each end of the path p t and t b t or simply t b the tree formed from p t 2 by attaching 1 p 2 s on an end of the p t 2 and 2 p 2 s on the vertex next to the end in li et al 2009 16 proved that among trees of order n with two vertices of maximum degree the maximal energy tree is either the graph t a or the graph t b where t n 4 4 3 however they could not determine which one of t a and t b is the maximal energy tree this is because the quasi order method is invalid for comparing their energies in this paper we use a new method to determine the maximal energy tree it turns out that things are more complicated we prove that the maximal energy tree is t b for 7 and any t 3 while the maximal energy tree is t a for 3 and any t 3 moreover for 4 the maximal energy tree is t a for all t 3 but one exception that t 4 for which t b is the maximal energy tree for 5 the maximal energy tree is t b for all t 3 but 44 exceptions that t is both odd and 3 t 89 for which t a is the maximal energy tree for 6 the maximal energy tree is t b for all t 3 but three exceptions that t 3 5 7 for which t a is the maximal energy tree one can see that for most cases of t b is the maximal energy tree 5 is a turning point and 3 and 4 are exceptional cases which means that for all chemical trees whose maximum degrees are at most 4 with two vertices of maximum degree at least 3 t a has maximal energy with only one exception t a 4 4

bicyclic digraphs with maximal energy if d is a digraph with n vertices then the energy of d is defined as e d k 1 n re z k where re z 1 re zn are the real parts of the eigenvalues z 1 z n of d in this paper we solve a problem proposed in khan et al 2015 we find the maximal value of the energy over the set of all bicyclic digraphs b n with n vertices

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

bicyclic signed graphs with minimal and second minimal energy abstract let s g be a signed graph of order n and size m where g is its underlying graph and is the sign function a connected signed graph s is said to be bicyclic if m n 1 in this paper we provide characterizations of bicyclic signed graphs with minimal and second minimal energy we notice that there are gaps in the proofs of the main results in j zhang h kan 2014 32 and j zhang b zhou 2005 31 which deal with minimal energy of bicyclic graphs and we plug these gaps as a consequence of the results obtained for bicyclic signed graphs

zero forcing sets and the minimum rank of graphs the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

quantum morphing and the jones polynomial we will explore the experimental observation that on the set of knots with bounded crossing number algebraically independent vassiliev invariants become correlated as noticed first by s willerton we will see this through the value distribution of the jones polynomial at roots of unit as the degree of the roots of unit is getting larger the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the jones polynomial we call such a phenomenon quantum morphing evaluations of the jones polynomial at roots of unity play a crucial role for example in the volume conjecture

the inverse inertia problem for the complements of partial k trees let f be an infinite field with characteristic not equal to two for a graph g v e with v 1 n let s g f be the set of all symmetric n n matrices a ai j over f with ai j 0 i j if and only if ij e we show that if g is the complement of a partial k tree and m k 2 then for all nonsingular symmetric m m matrices k over f there exists an m n matrix u such that utku s g f as a corollary we obtain that if k 2 m n and g is the complement of a partial k tree then for any two nonnegative integers p and q with p q m there exists a matrix in s g r with p positive and q negative eigenvalues

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

bounds for the sum balaban index and revised szeged index of regular graphs mathematical properties of many topological indices are investigated knor et al gave an upper bound for the balaban index of r regular graphs on n vertices and a better upper bound for fullerene graphs they also suggested exploring similar bounds for other topological indices in this paper we consider the sum balaban index and the revised szeged index and give upper and lower bounds for these three indices of r regular graphs and also the cubic graphs and fullerene graphs respectively

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

the spectral radius of edge chromatic critical graphs abstract a connected graph g with maximum degree and edge chromatic number g 1 is called critical if g e for every edge e of g in this paper we consider two weaker versions of vizing s conjecture which concern the spectral radius g and the signless laplacian spectral radius g of g we obtain some lower bounds for g and g and present some cases where the conjectures are true finally several open problems are also proposed

the two eigenvalue problem and density of jones representation of braid groups 1 the two eigenvalue problem 179 2 hecke algebra representations of braid groups 183 3 duality of jones wenzl representations 186 4 closed images of jones wenzl sectors 188 5 distribution of evaluations of jones polynomials 192 6 fibonacci representations 196

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

the bipartite unicyclic graphs with the first n 3 4 largest matching energies the theory of matching energy of graphs since been proposed by gutman and wagner in 2012 has attracted more and more attention it is denoted by b n m the class of bipartite graphs with order n and size m in particular b n n denotes the set of bipartite unicyclic graphs which is an interesting class of graphs in this paper for odd n we characterize the bipartite unicyclic graphs with the first n 3 4 largest matching energies there is an interesting correspondence we conclude that the graph with the second maximal matching energy in b n n for odd n 11 is p n 6 which is the only graph attaining the maximum value of the energy among all the bipartite unicyclic graphs for n 16

orthogonal representations minimum rank and graph complements orthogonal representations are used to show that complements of certain sparse graphs have positive semidefinite minimum rank at most 4 this bound applies to the complement of a 2 tree and to the complement of a unicyclic graph hence for such graphs the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph the minimum rank of the complement of a 2 tree is determined exactly

three connected graphs whose maximum nullity is at most three for a graph g v e with vertex set v 1 2 n let s g be the set of all n n real valued symmetric matrices a which represent g the maximum nullity of a graph g denoted by m g is the largest possible nullity of any matrix a s g fiedler showed that a graph g has m g 1 if and only if g is a path johnson et al gave a characterization of all graphs g with m g 2 independently hogben and van der holst gave a characterization of all 2 connected graphs with m g 2 in this paper we show that k connected graphs g have m g k that k connected partial k graphs g have m g k and that for 3 connected graphs g m g 3 if and only if g is a partial 3 path

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

extremal values of matching energies of one class of graphs in 1978 gutman proposed the concept of graph energy defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph which is related to the energy of electrons in conjugated hydrocarbons recently gutman and wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s in this paper we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies

the minimum semidefinite rank of the complement of partial k trees abstract for a graph g v e with v 1 n let s g be the set of all real symmetric n n matrices a a i j with a i j 0 i j if and only if ij e we prove the following results if g is the complement of a partial k tree h then there exists a positive semidefinite matrix a s g with rank a k 2 if in addition k 3 or g is k connected then there exist positive semidefinite matrices a s g and b s h such that rank a rank b n 2

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

note on unicyclic graphs with given number of pendent vertices and minimal energy abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let g n p denote the set of unicyclic graphs with n vertices and p pendent vertices in h hua m wang unicyclic graphs with given number of pendent vertices and minimal energy linear algebra appl 426 2007 478 489 hua and wang discussed the graphs that have minimal energy in g n p and determined the minimal energy graphs among almost all different cases of n and p in their work certain case of the values was left as an open problem in which the minimal energy species have to be chosen in two candidate graphs but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials this paper aims at solving the problem completely by using the well known coulson integral formula

on unicyclic graphs with given number of pendent vertices and minimal energy abstract 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

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

some properties of graphs constructed from 2 designs abstract let d p b be any nontrivial nonsymmetric 2 v k design in this paper we construct a connected regular and bipartite graph say ig v b d from d some graphic properties of ig v b d are investigated in particular the results in the paper bridges the equivalency between edge transitivity of ig v b d and flag transitivity of d with the equivalency it is obtained that ig v b d is a semisymmetric graph as long as d is a nonsymmetric flag transitive 2 v k design