Transactions on Combinatorics
Volume:9 Issue: 3, Sep 2020

Let $G$ 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.

‎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.

n 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.

‎‎‎The 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.

Consider 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.