In Section 4.5, “Assertion 4-2 based on Theorem 1, 2, 3, 4: Two undirected and simple graphs G1 G2. The number of nodes for the two is same as n. edges m. There are no isolated in both graphs. vertex edge adjacency matrices generated V1 V2, E1 E2. sums rows/columns produce four sets an array aV1, bV2, aE1, bE1 If only if ∑n k=1 ak = bk (ak V1)2 (bk V2)2, then aV1 bV2 a permutation another set; ...

‎For a simple graph $G$‎, ‎the signless Laplacian Estrada index is defined as $SLEE(G)=sum^{n}_{i=1}e^{q^{}_i}$‎, ‎where $q^{}_1‎, ‎q^{}_2‎, ‎dots‎, ‎q^{}_n$ are the eigenvalues of the signless Laplacian matrix of $G$‎. ‎In this paper‎, ‎we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$'s and then determine the unique unicyclic graph with maximum $SLEE$ a...

Let $G=(V,E)$, $V={v_1,v_2,ldots,v_n}$, be a simple graph with$n$ vertices, $m$ edges and a sequence of vertex degrees$Delta=d_1ge d_2ge cdots ge d_n=delta$, $d_i=d(v_i)$. Ifvertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $isim j$, otherwise, we write $insim j$. The first Zagreb index isvertex-degree-based graph invariant defined as$M_1(G)=sum_{i=1}^nd_i^2$, whereas the first Zag...

The causal set program and the Wolfram physics project leave open problem of how a graph that is (3+1)-dimensional Minkowski spacetime according to its simple geodesic distances could be generated solely from deterministic rules. This paper provides solution by describing rules characterize discrete Lorentz boosts between 4D lattice graphs, which combine further form Wigner rotations produce is...