University of IsfahanTransactions on Combinatorics2251-86579320200901The distance spectrum of two new operations of graphs1251382446710.22108/toc.2020.116372.1634ENZikaiTangKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaRenfangWuKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaHanlinChenKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaHanyuanDengKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaJournal Article20190409Let $G$ be a connected graph with vertex set $V(G)={v_1, v_2,ldots,v_n}$. The distance matrix $D=D(G)$ of $G$ is defined so that its $(i,j)$-entry is equal to the distance $d_G(v_i,v_j)$ between the vertices $v_i$ and $v_j$ of $G$. The eigenvalues ${mu_1, mu_2,ldots,mu_n}$ of $D(G)$ are the $D$-eigenvalues of $G$ and form the distance spectrum or the $D$-spectrum of $G$, denoted by $Spec_D(G)$. In this paper, we introduce two new operations $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ on graphs $G_1$ and $G_2$, and describe the distance spectra of $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ of regular graphs $G_1$ and $G_2 $ in terms of their adjacency spectra. By using these results, we obtain some new integral adjacency spectrum graphs, integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy.http://toc.ui.ac.ir/article_24467_d2bacaa1da976e615698f1efa93b6605.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579320200901On clique values identities and mantel-type theorems1391462463410.22108/toc.2020.119553.1680ENHossein TeimooriFaalDepartment of
Mathematics and Computer Science,
Allameh Tabatabai University, Tehran, Iran.Journal Article20191008In 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.http://toc.ui.ac.ir/article_24634_9320334e5b81d3a3208117a9faddb211.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579320200901On quadrilaterals in the suborbital graphs of the normalizer1471592445410.22108/toc.2020.120019.1685ENSerkanKaderDepartment of Mathematics,
Faculty of Arts and Sciences, Niğde Ömer Halisdemir University, Niğde, Turkeyhttps://orcid.org/0000-0001-5482-5727Bahadır OzgurGulerDepartment of Mathematics, Karadeniz Technical University, Trabzon, TurkeyElifAkşitDepartment of Mathematics, Niğde Ömer Halisdemir University, Niğde, TurkeyJournal Article20191111n this paper, we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$. Conditions for being an edge are obtained as a main tool, then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated.http://toc.ui.ac.ir/article_24454_4120f5466b20fdf04f01aaa231ece989.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579320200901Hosoya index of tree structures1611692466310.22108/toc.2020.121874.1713ENRaminKazemiDepartment of Statistics, Imam Khomeini International University, Qazvin, IranAliBehtoeiDepartment of Pure Mathematics, Imam Khomeini International University, Qazvin, IranJournal Article20200228The Hosoya index, also known as the $Z$ index, of a graph is the total number of matchings in it. In this paper, we study the Hosoya index of the tree structures. Our aim is to give some results on $Z$ in terms of Fibonacci numbers in such structures. Also, the asymptotic normality of this index is given.http://toc.ui.ac.ir/article_24663_a7db688468c0b041742a6b5024dfadb2.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579320200901Exact bounds for (λ,n)–stable 0-1 matrices.1711802458510.22108/toc.2020.120320.1692ENTrevorBruenFaculté de Médecine et des sciences de la santé, Université de Sherbrooke, Sherbrooke, CanadaJournal Article20191216Consider a v × v (0, 1) matrix A with exactly k ones in each row and each column. A is (λ, n)–stable, if it does not contain any λ × n submatrix with exactly one 0. If A is (λ, n)–stable, λ, n ≥ 2, then under suitable conditions on A, v ≥ k k(n−1)+(λ−2) . The case n λ−2 of equality leads to new and substantive connections with block designs. The previous bound and characterization of (λ, 2)–stable matrices follows immediately as a special case.http://toc.ui.ac.ir/article_24585_b98c4f1e8d5aa62e2d43e35851d93e95.pdf