- Applied Mathematics, Matrix Determinant, Algebraic Combinatorics, Algebraic Graph Theory, Combinatorial Matrix Theory, Number Theory, and 14 moreElectrical and Computer Engineering (ECE), Philosophy, Finance, Ethics, Mathematics, Optimal Control, Fuzzy Logic Control, شبکه های عصبی, Combinatorics, Fuzzy Mathematics, Fuzzy Control, Neural Networks, Wireless Sensor Networks, and GRACEFUL TREESedit
- Martin Loebledit
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A complete subgraph of a given graph is called a clique. A clique Polynomial of a graph is a generating function of the number of cliques in G. A real root of the clique polynomial of a graph G is called a clique root of G. Hajiabolhassan... more
A complete subgraph of a given graph is called a clique. A clique Polynomial of a graph is a generating function of the number of cliques in G. A real root of the clique polynomial of a graph G is called a clique root of G. Hajiabolhassan and Mehrabadi showed that the clique polynomial of any simple graph has a clique root in [−1, 0). As a generalization of their result, the author of this paper showed that the class of K4-free connected chordal graphs has also only clique roots. A given graph G is called flat if each edge of G belongs to at most two triangles of G. In answering the author’s open question about the class of non-chordal graphs with the same property of having only c;ique roots, we extend the aforementioned result to the class of K4free flat graphs. In particular, we prove that the class of K4-free flat graphs without isolated edges has r = −1 as one of it’s clique roots. We finally present some interesting open questions and conjectures regarding clique roots of graphs.
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In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of Kadison-Singer , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real... more
In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of Kadison-Singer , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real root for sums of polynomials with (at least) one real root. Then, as an immediate application of this simple but potentially useful lemma we characterize several class of graphs which have only clique roots. Finally, we conclude our paper with several interesting open problems and conjectures for interested readers.
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In this paper we develop a new geometric method to answer the log-concavity questions related to a nice class of combinatorial sequences arising from the Khayyam-Pascal triangle. 1
In this paper, we first review the graph-theoretical interpretations of the determinant and cofactors of a matrix, using the idea of cycle covers of the associated digraph of that matrix. We then also review the multiset analogue of the... more
In this paper, we first review the graph-theoretical interpretations of the determinant and cofactors of a matrix, using the idea of cycle covers of the associated digraph of that matrix. We then also review the multiset analogue of the combinatorial interpretation of the determinant based on the idea of Lydon covers. As the main result of this paper, we also give a multiset analogue of the cofactor of any entry of a matrix by giving a generalization of the concept of the Lydon cover. We then obtain a multiset analogue of the well-known Cayley-Hamilton theorem, as an application of the main result of this paper. Finally, we conclude the paper with several interesting open problems and conjectures.
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In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate... more
In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate generalization of the Mantel's classical result which asks for the maximum number of edges in triangle-free graphs to the class of $K_{4}$-free graphs. Then, by defining the concept of value for cliques (complete subgraphs) of higher orders, we also extend the classical result of Mantel for any graph $G$. We finally conclude our paper with a discussion about the possible future works.
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The clique polynomial of a graph G is the ordinary generating function of the number of complete subgraphs (cliques) of $G$. In this paper, we introduce a new vertex-weighted version of these polynomials. We also show that these weighted... more
The clique polynomial of a graph G is the ordinary generating function of the number of complete subgraphs (cliques) of $G$. In this paper, we introduce a new vertex-weighted version of these polynomials. We also show that these weighted clique polynomials have always a real root provided that the weights are non-negative real numbers. As an application, we obtain a no-homomorphism criteria based on the largest real root of our vertex-weighted clique polynomial.
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In this paper, we give a generalization of Ibn al-Haytham recursive formula for sums of powers of any integer sequence. Then, we obtain higher dimensional generalizations of the generalized Ibn al-Haytham formula. As by-products, we also... more
In this paper, we give a generalization of Ibn al-Haytham recursive formula for sums of powers of any integer sequence. Then, we obtain higher dimensional generalizations of the generalized Ibn al-Haytham formula. As by-products, we also show that how our recursive formulas imply other interesting integer sequences identities like Karaji L-summing equation and Abel's summation by parts lemma. Finally, as an application, we prove several identities related to Fibonnaci and harmonic numbers.
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In this paper we will present a new method to calculate determinants of square matrices. The method is based on the Chio-Dodgson's condensation formula and our approach automatically affects in reducing the order of determinants by... more
In this paper we will present a new method to calculate determinants of square matrices. The method is based on the Chio-Dodgson's condensation formula and our approach automatically affects in reducing the order of determinants by two. Also, using the Chio's condensation method we present an inductive proof of Dodgson's determinantal identity.
In this paper, we present a multiset analogue of the even-odd permutations identity in the context of combinatorics of words. The multiset version is indeed equivalent to the coin arrangements lemma which is a key lemma in Sherman’s proof... more
In this paper, we present a multiset analogue of the even-odd permutations identity in the context of combinatorics of words. The multiset version is indeed equivalent to the coin arrangements lemma which is a key lemma in Sherman’s proof of Feynman’s conjecture about combinatorial solution of Ising model in statistical physics. Here, we give a bijective proof which is based on the standard factorization of a Lyndon word.
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Abstract In this paper, we first give a multiset version of the graph-theoretical interpretation of the classic determinant of a matrix A based on a multiset generalization of the cycle cover of its associated digraph D ( A ) . Then, as a... more
Abstract In this paper, we first give a multiset version of the graph-theoretical interpretation of the classic determinant of a matrix A based on a multiset generalization of the cycle cover of its associated digraph D ( A ) . Then, as a direct consequence of this interpretation, we present another algebraic proof of a weighted version of the original coin arrangements lemma.
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In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Mason’s theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation)... more
In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Mason’s theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) $$ a_{1}^{m_1} + a_{2}^{m_2} + \cdots + a_{n-1}^{m_{n-1}} = a_{n}^{m_{n}}, $$ where $ a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}[x_{1}, \ldots, x_{l}] $ , such that k out of the n-polynomials $ (k \leq n-2) $ are constant, and $ m_{1}, m_{2}, \ldots, m_{n} \in \mathbb{N}, $ under certain conditions for $ a_{i}(i = 1, \ldots, n) $, has no non-constant solution.
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In this talk, we first briefly review some important combinatorial and algebraic properties of the generating function of i-cliques in a given graph G. Here, by an i-clique we mean a complete subgraph of G on i vertices. We will call the... more
In this talk, we first briefly review some important combinatorial and algebraic properties of the generating function of i-cliques in a given graph G. Here, by an i-clique we mean a complete subgraph of G on i vertices. We will call the above generating function the clique ...