trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

treediagonal matrices and their inverses abstract a generalization of tridiagonal matrices is considered namely treediagonal matrices which have nonzero off diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements some properties of treediagonal matrices are given and their inverses are characterized and shown to have an interesting structure

on the laplacian coefficients of tricyclic graphs abstract let g d e t i n l g k 0 n 1 k c k n k be the characteristic polynomial of the laplacian matrix of a graph g of order n in this paper we show that among all connected tricyclic graphs of order n the k th coefficient c k is smallest when the graph is b n 7 1 3 3 3 obtained from the complete graph k 4 by adding n 4 pendent vertices attached to the vertex of degree 3 and for some lemmas in c x he h y shan on the laplacian coefficients of bicyclic graphs discrete math 310 2010 3404 3412 we present a new method to prove them

the signless laplacian coefficients and the incidence energy of the graphs without even cycles abstract let n m be the set of all connected graphs of order n with m edges without even cycles where n m 3 2 n 1 we consider the signless laplacian coefficients and the incidence energy in n m two new graph transformations are proposed here among n m it is obtained that there is exactly one minimal element with respect to the quasi ordering according to their signless laplacian coefficients furthermore the graph with the minimal incidence energy is found among n m

the coefficients of laplacian characteristic polynomials of graphs abstract li et al j x li w c shiu a chang the number of spanning trees of a graph appl math lett 23 2010 286 290 obtained some upper bounds for the number of spanning trees of graphs in this paper we characterize the connected graph g with the connectivity resp edge connectivity and chromatic number which has the maximal coefficients of the laplacian characteristic polynomial since the number of spanning trees of a graph g is determined by one of the coefficients of the laplacian characteristic polynomial of g we generalize the results by li et al

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

laplacian coefficients of trees with a given bipartition let g be a graph of order n and g k 0n 1 kck n k the laplacian characteristic polynomial of g zhou and gutman 19 proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for a star in this paper for two given positive integers p and q p q we characterize the trees with a given bipartition p q which have the minimal and second minimal laplacian coefficients

on the laplacian coefficients of signed graphs abstract let g be a signed graph where g is its underlying graph and its sign function defined on edges of g a signed graph the subgraph of is its signed tu subgraph if the signed graph induced by the vertices of consists of trees and or unbalanced unicyclic signed graphs let l d g a be the laplacian of in this paper we express the coefficient of the laplacian characteristic polynomial of based on the signed tu subgraphs of and establish the relation between the laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph as an application we identify the signed unicyclic graphs having extremal coefficients of the laplacian characteristic polynomial

on the ordering of trees by the laplacian coefficients abstract we generalize the results from x d zhang x p lv y h chen ordering trees by the laplacian coefficients linear algebra appl 2009 doi 10 1016 j laa 2009 04 018 on the partial ordering of trees with given diameter for two n vertex trees t 1 and t 2 if c k t 1 c k t 2 holds for all laplacian coefficients c k k 0 1 n we say that t 1 is dominated by t 2 and write t 1 c t 2 we proved that among n vertex trees with fixed diameter d the caterpillar c n d has minimal laplacian coefficients c k k 0 1 n the number of incomparable pairs of trees on 18 vertices is presented as well as infinite families of examples for two other partial orderings of trees recently proposed by mohar for every integer n we construct a chain t i i 0 m of n vertex trees of length n 2 4 such that t 0 s n t m p n and t i c t i 1 for all i 0 1 m 1 in addition the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths we determine the relations among the extremal trees with fixed maximum degree and with perfect matching and further support the laplacian coefficients as a measure of branching

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

on the laplacian coefficients of acyclic graphs abstract let g be a graph of order n and let g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix zhou and gutman recently proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for stars a new proof and a strengthening of this result is provided a relation to the wiener index is discussed

on laplacian energy like invariant of a graph abstract let g be a simple graph of order n and let 1 2 n 0 be the laplacian spectrum of g the laplacian energy like invariant of g lel for short is defined as lel g i 1 n 1 i in this paper a new lower bound for lel of graphs in terms of the maximum degree is given meanwhile an upper bound and a lower bound for lel of the line graph resp the subdivision graph and the total graph of a regular graph g are obtained

determinantal formulae for matrices with sparse inverses abstract the determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be read off from the graph of the inverse of the matrix the only information used is the zero pattern of the inverse and each zero pattern yields one or more corresponding formulae for the determinant

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

invertible completions of partial operator matrices the nonsymmetric case abstract we consider nonsymmetric partial operator matrices r whose directed graphs belong to a certain class and for which certain key fully specified principal submatrices are invertible under these circumstances we prove the existence of a unique invertible completion f of r such that f 1 ij is block zero whenever rij is unspecified in this way the existing results of johnson and lundquist are generalized to the nonsymmetric case

positive p matrices completion problem with zeros in the inverse completion in this paper based on the completion problem of partial p matrix for 1 chordal graph we obtain the partial positive p matrix completions of k chordal graphs and k chordal block graphs under some conditions and solve partial positive p matrix completion problems with zeros in the inverse completion for k chordal graphs and k chordal block graphs k 2

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

on the laplacian coefficients of graphs under some transformations abstract we generalize three approaches on graph transformations respectively from stevanovic and ilic 2009 16 and tan 2011 19 we also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the laplacian coefficients of graphs from li and feng 1979 26 moreover we determine the unique tree having the third maximal laplacian coefficients among all n vertex trees

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

on the signless laplacian coefficients of unicyclic graphs let g be a graph of order n and let qg x i 0n 1 ipi g xn i be the characteristic polynomial of the signless laplacian of g let eg n respectively cg sn g 1 denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle cg with an end vertex respectively the center of the path pn g 1 respectively the star sn g 1 it is proved that for k 2 n 1 as g varies over all unicyclic graphs of order n depending on k and n the maximum value of pk g is attained at g cn or e3 n and the minimum value is attained uniquely at g c4 sn 3 or c3 sn 2 except for the resolution of a conjecture on cubic polynomials the uniqueness issue for the maximization problem is also settled

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

ordering of trees with fixed matching number by the laplacian coefficients abstract let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g x k 0 n 1 k c k x n k aleksandar ilic a ilic trees with minimal laplacian coefficients comput math appl 59 2010 2776 2783 identified n vertex trees with given matching number q which simultaneously minimize all laplacian coefficients in this paper we give another proof of this result generalizing the approach in the above paper we determine n vertex trees with given matching number q which have the second minimal laplacian coefficients we also identify the n vertex trees with a perfect matching having the largest and the second largest laplacian coefficients respectively extremal values on some indices such as wiener index modified hyper wiener index laplacian like energy incidence energy of n vertex trees with matching number q are obtained in this paper

on the norm of the nilpotent residuals of all subgroups of a finite group abstract r baer and wielandt in 1934 and 1958 respectively considered the intersection of the normalizers of all subgroups of g and the intersection of the normalizers of all subnormal subgroups of g in this paper for a finite group g we define the subgroup s g to be the intersection of the normalizers of the nilpotent residuals of all subgroups of g set s 0 1 define s i 1 g s i g s g s i g for i 1 by s g denote the terminal term of the ascending series it is proved that g s g if and only if the nilpotent residual g n is nilpotent furthermore if all elements of prime order of g are in s g then g is solvable and l p g 1 where l p g is p length of g for p g where g means the set of prime divisors of g